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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153501.png" />''
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$#C+1 = 386 : ~/encyclopedia/old_files/data/B015/B.0105350 Basis
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A minimal subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153502.png" /> that generates it. Generation here means that by application of operations of a certain class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153503.png" /> to elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153504.png" /> it is possible to obtain any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153505.png" />. This concept is related to the concept of dependence: By means of operations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153506.png" /> the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153507.png" /> become dependent on the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153508.png" />. Minimality means that no proper subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b0153509.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535010.png" />. In a certain sense this property causes the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535011.png" /> to be independent: None of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535012.png" /> is generated by the other elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535013.png" />. For instance, the set of all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535014.png" /> has the unique element 0 as basis and is generated from it by the operation of immediate succession and its iteration. The set of all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535015.png" /> is generated by the operation of multiplication from the basis consisting of all prime numbers. A basis of the algebra of quaternions consists of the four elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535016.png" /> if the generating operations consist of addition and of multiplication by real numbers; if, in addition to these operations, one also includes multiplication of quaternions, the basis will consist of three elements only — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535017.png" /> (because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535018.png" />).
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A basis of the natural numbers of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535020.png" /> is a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535021.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535022.png" /> of natural numbers including 0, which, as a result of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535023.png" />-fold addition to itself (the generating operation) yields all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535024.png" />. This means that any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535025.png" /> can be represented in the form
+
''of a set $  X $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535026.png" /></td> </tr></table>
+
A minimal subset  $  B $
 +
that generates it. Generation here means that by application of operations of a certain class $  \Omega $
 +
to elements  $  b \in B $
 +
it is possible to obtain any element  $  x \in X $.
 +
This concept is related to the concept of dependence: By means of operations from  $  \Omega $
 +
the elements of  $  X $
 +
become dependent on the elements of  $  B $.
 +
Minimality means that no proper subset  $  B _ {1} \subset  B $
 +
generates  $  X $.
 +
In a certain sense this property causes the elements of  $  B $
 +
to be independent: None of the elements  $  b \in B $
 +
is generated by the other elements of  $  B $.
 +
For instance, the set of all natural numbers  $  \mathbf Z _ {0} $
 +
has the unique element 0 as basis and is generated from it by the operation of immediate succession and its iteration. The set of all natural numbers  $  >1 $
 +
is generated by the operation of multiplication from the basis consisting of all prime numbers. A basis of the algebra of quaternions consists of the four elements  $  \{ 1, i, j, k \} $
 +
if the generating operations consist of addition and of multiplication by real numbers; if, in addition to these operations, one also includes multiplication of quaternions, the basis will consist of three elements only —  $  \{ 1, i, j \} $(
 +
because  $  k=ij $).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535027.png" />. For example, every natural number is a sum of four squares of natural numbers (Lagrange's theorem), i.e. the sequence of squares is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535028.png" /> of order 4. In general, the sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535029.png" />-th powers of natural numbers is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535030.png" /> (Hilbert's theorem), the order of which has been estimated by the [[Vinogradov method|Vinogradov method]]. The concept of a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535031.png" /> has been generalized to the case of arbitrary sequences of numbers, i.e. functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535032.png" />.
+
A basis of the natural numbers of order  $  k $
 +
is a subsequence  $  \Omega $
 +
of the set  $  \mathbf Z _ {0} $
 +
of natural numbers including 0, which, as a result of $  k $-
 +
fold addition to itself (the generating operation) yields all of $  \mathbf Z _ {0} $.  
 +
This means that any natural number  $  n $
 +
can be represented in the form
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535033.png" /> always contains a generating set (in the trivial case: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535034.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535035.png" />), but minimality may prove to be principally impossible (such a situation is typical of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535036.png" /> containing infinite-place operations, in particular in topological structures, lattices, etc.). For this reason the minimality condition is replaced by a weaker requirement: A basis is a generating set of minimal cardinality. In this context a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535037.png" /> is defined as a parametrized set (or population), i.e. as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535038.png" /> on a set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535039.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535040.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535041.png" />; the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535042.png" /> is sometimes called as the dimension (or rank) of the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535043.png" />. For example, a countable everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535044.png" /> in a separable topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535045.png" /> may be considered as a basis for it; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535046.png" /> is generated from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535047.png" /> by the closure operation (which, incidentally, is related to generation in more general cases as well, see below).
+
$$
 +
= a _ {1} + \dots + a _ {k} ,
 +
$$
  
A basis for a topology of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535048.png" /> (a [[Base|base]]) is a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535049.png" /> of the set of all open subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535050.png" />; the generation is effected by taking unions of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535051.png" />.
+
where  $  a _ {i} \in \Omega $.
 +
For example, every natural number is a sum of four squares of natural numbers (Lagrange's theorem), i.e. the sequence of squares is a basis of  $  \mathbf Z _ {0} $
 +
of order 4. In general, the sequence of  $  m $-
 +
th powers of natural numbers is a basis of  $  \mathbf Z _ {0} $(
 +
Hilbert's theorem), the order of which has been estimated by the [[Vinogradov method|Vinogradov method]]. The concept of a basis of $  \mathbf Z _ {0} $
 +
has been generalized to the case of arbitrary sequences of numbers, i.e. functions on  $  \mathbf Z _ {0} $.
  
A basis of a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535052.png" /> (a dual base of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535053.png" /> in the sense of Tarski) is a dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535054.png" /> (of minimal cardinality) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535055.png" />; the generation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535056.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535057.png" /> (and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535058.png" /> itself) is determined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535059.png" /> (which is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535060.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535063.png" /> is the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535064.png" /> and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535065.png" /> as a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535066.png" /> such that for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535067.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535068.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535069.png" />.
+
A set  $  X $
 +
always contains a generating set (in the trivial case: $  X $
 +
generates  $  X $),
 +
but minimality may prove to be principally impossible (such a situation is typical of classes  $  \Omega $
 +
containing infinite-place operations, in particular in topological structures, lattices, etc.). For this reason the minimality condition is replaced by a weaker requirement: A basis is a generating set of minimal cardinality. In this context a basis  $  B $
 +
is defined as a parametrized set (or population), i.e. as a function  $  b(t) $
 +
on a set of indices  $  T $
 +
with values in  $  X $,  
 +
such that  $  b(T) = B  $;
 +
the cardinality of  $ T $
 +
is sometimes called as the dimension (or rank) of the basis of $  X $.  
 +
For example, a countable everywhere-dense set  $  B $
 +
in a separable topological space  $  P $
 +
may be considered as a basis for it;  $  P $
 +
is generated from  $  B $
 +
by the closure operation (which, incidentally, is related to generation in more general cases as well, see below).
  
More special cases of bases of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535070.png" /> are introduced according to the following procedure. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535071.png" /> be the Boolean algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535072.png" />, i.e. the set of all its subsets. A generating operator (or a closure operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535073.png" /> is a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535074.png" /> into itself such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535076.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535077.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535078.png" />.
+
A basis for a topology of a topological space  $  X $(
 +
a [[Base|base]]) is a basis  $  \mathfrak B $
 +
of the set of all open subsets in  $  X $;
 +
the generation is effected by taking unions of elements of $  \mathfrak B $.
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535079.png" /> is generated by a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535080.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535081.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535082.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535083.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535084.png" />. A minimal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535085.png" /> possessing this property is said to be a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535086.png" /> defined by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535087.png" />. A generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535088.png" /> is of finite type if, for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535090.png" />, it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535091.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535092.png" /> for a certain finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535093.png" />; a generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535094.png" /> has the property of substitution if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535096.png" />, both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535098.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b01535099.png" />. A generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350100.png" /> of finite type with the substitution property defines a dependence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350101.png" />, i.e. a subdivision of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350102.png" /> into two classes — dependent and independent sets; a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350103.png" /> is said to be dependent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350104.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350105.png" />, and is said to be independent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350106.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350107.png" />. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350108.png" /> is dependent (independent) if and only if some (arbitrary) non-empty finite subset(s) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350109.png" /> is dependent (are independent).
+
A basis of a Boolean algebra  $  \mathfrak A $(
 +
a dual base of  $  \mathfrak A $
 +
in the sense of Tarski) is a dense set $  S $(
 +
of minimal cardinality) in  $  \mathfrak A $;  
 +
the generation of  $  \mathfrak A $
 +
from  $  S $(
 +
and hence  $  S $
 +
itself) is determined by the condition  $  s \rightarrow a = \lor $(
 +
which is equivalent to  $  s \subset  a $),  
 +
where  $  s \in S $,  
 +
a \in \mathfrak A $,
 +
$  \lor $
 +
is the unit of $  \mathfrak A $
 +
and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter  $  \nabla $
 +
as a set $  S $
 +
such that for an arbitrary $  a \in \nabla $
 +
there exists an  $  s \in S $
 +
with  $  s \subset a $.
  
For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350110.png" /> to be a basis of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350111.png" /> it is necessary and sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350112.png" /> to be an independent generating set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350113.png" />, or else, a maximal independent set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350114.png" />.
+
More special cases of bases of a set $  X $
 +
are introduced according to the following procedure. Let  $  B(X) $
 +
be the Boolean algebra of  $  X $,
 +
i.e. the set of all its subsets. A generating operator (or a closure operator)  $  J $
 +
is a mapping of  $  B (X) $
 +
into itself such that if  $  A \subset  B $,  
 +
then  $  J(A) \subset  J(B) $;
 +
$  A \subset  J(A) $;
 +
$  JJ(A) = J(A) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350115.png" /> is an arbitrary independent set, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350116.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350117.png" />-generating set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350118.png" />, then there exists a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350119.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350120.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350121.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350122.png" /> always has a basis, and any two bases of it have the same cardinality.
+
An element  $  x \in X $
 +
is generated by a set $  A $
 +
if  $  x \in J(A) $;
 +
in particular, $  A $
 +
generates  $  X $
 +
if  $  J(A) = X $.  
 +
A minimal set  $  B $
 +
possessing this property is said to be a basis of  $  X $
 +
defined by the operator  $  J $.  
 +
A generating operator  $  J $
 +
is of finite type if, for arbitrary  $  A \subset  X $
 +
and  $  x \subset  X $,
 +
it follows from  $  x \in J(A) $
 +
that  $  x \in J(A _ {0} ) $
 +
for a certain finite subset  $  A _ {0} \subset  A $;
 +
a generating operator  $  J $
 +
has the property of substitution if, for any  $  y, z \in X $
 +
and  $  A \subset  X $,
 +
both  $  y \notin J(A) $
 +
and  $  y \in J(A \cup \{ z \} ) $
 +
imply that $  z \in J(A \cup \{ y \} ) $.  
 +
A generating operator  $  J $
 +
of finite type with the substitution property defines a dependence relation on  $  X $,  
 +
i.e. a subdivision of  $  B(X) $
 +
into two classes — dependent and independent sets; a set  $  A $
 +
is said to be dependent if  $  y \in J(A \setminus  y) $
 +
for some  $  y \in A $,  
 +
and is said to be independent if  $  y \notin J (A \setminus  y) $
 +
for any $  y \in A $.
 +
Therefore,  $  A $
 +
is dependent (independent) if and only if some (arbitrary) non-empty finite subset(s)  $  A _ {0} \subset  A $
 +
is dependent (are independent).
  
In algebraic systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350123.png" /> an important role is played by the concept of the so-called free basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350124.png" />, which is characterized by the following property: Any mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350125.png" /> into any algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350126.png" /> (of the same signature) may be extended to a (unique) (homo)morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350127.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350128.png" /> or, which is the same thing, for any (homo)morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350129.png" /> and any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350130.png" />, the generating operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350132.png" /> satisfy the condition:
+
For a set  $  B $
 +
to be a basis of the set  $  X $
 +
it is necessary and sufficient for  $  B $
 +
to be an independent generating set for  $  X $,
 +
or else, a maximal independent set in  $  X $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350133.png" /></td> </tr></table>
+
If  $  A $
 +
is an arbitrary independent set, and  $  C $
 +
is an  $  X $-
 +
generating set containing  $  A $,
 +
then there exists a basis  $  B $
 +
in  $  X $
 +
such that  $  A \subset  B \subset  C $.
 +
In particular,  $  X $
 +
always has a basis, and any two bases of it have the same cardinality.
 +
 
 +
In algebraic systems  $  X $
 +
an important role is played by the concept of the so-called free basis  $  B $,
 +
which is characterized by the following property: Any mapping of  $  B \subset  X $
 +
into any algebraic system  $  Y $(
 +
of the same signature) may be extended to a (unique) (homo)morphism from  $  X $
 +
into  $  Y $
 +
or, which is the same thing, for any (homo)morphism  $  \theta : X \rightarrow Y $
 +
and any set  $  A \subset  X $,
 +
the generating operators  $  J _ {X} $
 +
and  $  J _ {Y} $
 +
satisfy the condition:
 +
 
 +
$$
 +
\theta \{ J _ {X} (A) \}  = \
 +
J _ {Y} ( \theta \{ A \} ) .
 +
$$
  
 
An algebraic system with a free basis is said to be free.
 
An algebraic system with a free basis is said to be free.
  
A typical example is a basis of a (unitary) module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350134.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350135.png" />, that is, a free family of elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350136.png" /> generating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350137.png" /> [[#References|[3]]]. Here, a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350138.png" /> of elements of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350139.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350140.png" /> is said to be free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350141.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350142.png" /> for all except a finite number of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350143.png" />) implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350144.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350145.png" />, and the generation is realized by representing the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350146.png" /> as linear combinations of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350147.png" />: There exists a set (dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350148.png" />) of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350149.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350150.png" /> for all except a finite number of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350151.png" />, and such that the decomposition
+
A typical example is a basis of a (unitary) module $  M $
 +
over a ring $  K $,  
 +
that is, a free family of elements from $  M $
 +
generating $  M $[[#References|[3]]]. Here, a family $  A = \{ {a _ {t} } : {t \in T } \} $
 +
of elements of a $  K $-
 +
module $  M $
 +
is said to be free if $  \sum \xi _ {t} a _ {t} = 0 $(
 +
where $  \xi _ {t} = 0 $
 +
for all except a finite number of indices $  t $)  
 +
implies that $  \xi _ {t} = 0 $
 +
for all $  t $,  
 +
and the generation is realized by representing the elements $  x $
 +
as linear combinations of the elements $  a _ {t} $:  
 +
There exists a set (dependent on $  x $)  
 +
of elements $  \xi _ {t} \in K $
 +
such that $  \xi _ {t} = 0 $
 +
for all except a finite number of indices $  t $,  
 +
and such that the decomposition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350152.png" /></td> </tr></table>
+
$$
 +
= \sum \xi _ {t} a _ {t}  $$
  
is valid (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350153.png" /> is the linear envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350154.png" />). In this sense, the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350155.png" /> is free basis; the converse proposition is also true. Thus, the set of periods of a doubly-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350156.png" /> of one complex variable, which is a discrete Abelian group (and hence a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350157.png" />), has a free basis, called the period basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350158.png" />; it consists of two so-called primitive periods. A period basis of an Abelian function of several complex variables is defined in a similar manner.
+
is valid (i.e. $  X $
 +
is the linear envelope of $  A $).  
 +
In this sense, the basis $  M $
 +
is free basis; the converse proposition is also true. Thus, the set of periods of a doubly-periodic function $  f $
 +
of one complex variable, which is a discrete Abelian group (and hence a module over the ring $  \mathbf Z $),  
 +
has a free basis, called the period basis of $  f $;  
 +
it consists of two so-called primitive periods. A period basis of an Abelian function of several complex variables is defined in a similar manner.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350159.png" /> is a skew-field, all bases (in the previous sense) are free. On the contrary, there exist modules without a free basis; these include, for example, the non-principal ideals in an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350160.png" />, considered as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350161.png" />-module.
+
If $  K $
 +
is a skew-field, all bases (in the previous sense) are free. On the contrary, there exist modules without a free basis; these include, for example, the non-principal ideals in an integral domain $  K $,  
 +
considered as a $  K $-
 +
module.
  
A basis of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350162.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350163.png" /> is a (free) basis of the unitary module which underlies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350164.png" />. In a similar manner, a basis of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350165.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350166.png" /> is a basis of the vector space underlying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350167.png" />. All bases of a given vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350168.png" /> have the same cardinality, which is equal to the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350169.png" />; the latter is called the algebraic dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350170.png" />. Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350171.png" /> can be represented as a linear combination of basis elements in a unique way. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350172.png" />, which are linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350173.png" />, are called the components (coordinates) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350174.png" /> in the given basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350175.png" />.
+
A basis of a vector space $  X $
 +
over a field $  K $
 +
is a (free) basis of the unitary module which underlies $  X $.  
 +
In a similar manner, a basis of an algebra $  A $
 +
over a field $  K $
 +
is a basis of the vector space underlying $  A $.  
 +
All bases of a given vector space $  X $
 +
have the same cardinality, which is equal to the cardinality of $  T $;  
 +
the latter is called the algebraic dimension of $  X $.  
 +
Each element $  x \in X $
 +
can be represented as a linear combination of basis elements in a unique way. The elements $  \xi _ {t} (x) \in K $,  
 +
which are linear functionals on $  X $,  
 +
are called the components (coordinates) of $  x $
 +
in the given basis $  \{ a _ {t} \} $.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350176.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350177.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350178.png" /> is a maximal (with respect to inclusion) [[Free set|free set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350179.png" />.
+
A set $  A $
 +
is a basis in $  X $
 +
if and only if $  A $
 +
is a maximal (with respect to inclusion) [[Free set|free set]] in $  X $.
  
 
The mapping
 
The mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350180.png" /></td> </tr></table>
+
$$
 +
\Xi : x  \rightarrow  \xi _ {x} (t),
 +
$$
 +
 
 +
where  $  \xi _ {x} (t) = \xi _ {t} (x) $
 +
if  $  \xi _ {t} $
 +
is the value of the  $  t $-
 +
th component of  $  x $
 +
in the basis  $  A $,
 +
and 0 otherwise, is called the basis mapping; it is a linear injective mapping of  $  X $
 +
into the space  $  K  ^ {T} $
 +
of functions on  $  T $
 +
with values in  $  K $.
 +
In this case the image  $  \Xi (X) $
 +
consists of all functions with a finite number of non-zero values (functions of finite support). This interpretation permits one to define a generalized basis of a vector space  $  X $
 +
over a field  $  K $
 +
as a bijective linear mapping from it to some subspace  $  K (T) $
 +
of the space  $  K  ^ {T} $
 +
of functions on  $  T $
 +
with values in  $  K $,
 +
where  $  T $
 +
is some suitably chosen set. However, unless additional restrictions (e.g. an order) and additional structures (e.g. a topology) are imposed on  $  T $,
 +
and corresponding compatible conditions on  $  K(T) $
 +
are introduced, the concept of a generalized basis is seldom of use in practice.
 +
 
 +
A basis of a vector space  $  X $
 +
is sometimes called an algebraic basis; in this way it is stressed that there is no connection with additional structures on  $  X $,
 +
even if they are compatible with its vector structure.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350181.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350182.png" /> is the value of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350183.png" />-th component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350184.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350185.png" />, and 0 otherwise, is called the basis mapping; it is a linear injective mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350186.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350187.png" /> of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350188.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350189.png" />. In this case the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350190.png" /> consists of all functions with a finite number of non-zero values (functions of finite support). This interpretation permits one to define a generalized basis of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350191.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350192.png" /> as a bijective linear mapping from it to some subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350193.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350194.png" /> of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350195.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350196.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350197.png" /> is some suitably chosen set. However, unless additional restrictions (e.g. an order) and additional structures (e.g. a topology) are imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350198.png" />, and corresponding compatible conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350199.png" /> are introduced, the concept of a generalized basis is seldom of use in practice.
+
A Hamel basis is a basis of the field of real numbers  $  \mathbf R $,
 +
considered as a vector space over the field of rational numbers. It was introduced by G. Hamel [[#References|[4]]] to obtain a discontinuous solution of the functional equation  $  f(x+y) = f(x) + f(y) $;
 +
the graph of its solution is everywhere dense in the plane  $  \mathbf R  ^ {2} $.  
 +
To each almost-periodic function corresponds some countable Hamel basis  $  \beta $
 +
such that each Fourier exponent  $  \Lambda _ {n} $
 +
of this function belongs to the linear envelope of  $  \beta $.  
 +
The elements of $  \beta $
 +
may be so chosen that they belong to a sequence  $  \{ \Lambda _ {i} \} $;
 +
the set  $  \beta $
 +
is said to be a basis of the almost-periodic functions. An analogous basis has been constructed in a ring containing a skew-field $  P $
 +
and which has the unit of  $  P $
 +
as its own unit. An algebraic basis of an arbitrary vector space is also sometimes referred to as a Hamel basis.
  
A basis of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350200.png" /> is sometimes called an algebraic basis; in this way it is stressed that there is no connection with additional structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350201.png" />, even if they are compatible with its vector structure.
+
A topological basis (a basis of a topological vector space $  X $
 +
over a field  $  K $)
 +
is a set  $  A = \{ {a _ {t} } : {t \in T } \} \subset  X $
 +
with properties and functions analogous to those of the algebraic basis of the vector space. The concept of a topological basis, which is one of the most important ones in functional analysis, generalizes the concept of an algebraic basis with regard to the topological structure of  $  X $
 +
and makes it possible to obtain, for each element  $  X $,
 +
its decomposition with respect to the basis  $  \{ a _ {t} \} $,
 +
which is moreover unique, i.e. a representation of  $  x $
 +
as a limit (in some sense) of linear combinations of elements  $  a _ {t} $:
  
A Hamel basis is a basis of the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350202.png" />, considered as a vector space over the field of rational numbers. It was introduced by G. Hamel [[#References|[4]]] to obtain a discontinuous solution of the functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350203.png" />; the graph of its solution is everywhere dense in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350204.png" />. To each almost-periodic function corresponds some countable Hamel basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350205.png" /> such that each Fourier exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350206.png" /> of this function belongs to the linear envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350207.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350208.png" /> may be so chosen that they belong to a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350209.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350210.png" /> is said to be a basis of the almost-periodic functions. An analogous basis has been constructed in a ring containing a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350211.png" /> and which has the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350212.png" /> as its own unit. An algebraic basis of an arbitrary vector space is also sometimes referred to as a Hamel basis.
+
$$
 +
x  =  \lim\limits  \sum \xi _ {t} (x)a _ {t} ,
 +
$$
  
A topological basis (a basis of a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350213.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350214.png" />) is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350215.png" /> with properties and functions analogous to those of the algebraic basis of the vector space. The concept of a topological basis, which is one of the most important ones in functional analysis, generalizes the concept of an algebraic basis with regard to the topological structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350216.png" /> and makes it possible to obtain, for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350217.png" />, its decomposition with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350218.png" />, which is moreover unique, i.e. a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350219.png" /> as a limit (in some sense) of linear combinations of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350220.png" />:
+
where  $  \xi _ {t} (x) $
 +
are linear functionals on  $  X $
 +
with values in  $  K $,
 +
called the components of $  x $
 +
in the basis $  A $,  
 +
or the coefficients of the decomposition of $  x $
 +
with respect to the basis  $  A $.
 +
Clearly, for the decomposition of an arbitrary  $  x $
 +
to exist,  $  A $
 +
must be a complete set in  $  X $,  
 +
and for such a decomposition to be unique (i.e. for the zero element of $  X $
 +
to have all components equal to zero),  $  A $
 +
must be a topologically free set in $  X $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350221.png" /></td> </tr></table>
+
The sense and the practical significance of a topological basis (which will be simply denoted as a  "basis" in what follows) is to establish a bijective linear mapping of  $  X $,
 +
called the basis mapping,  $  \Xi $
 +
into some (depending on  $  X $)
 +
space  $  K(T) $
 +
of functions with values in  $  K $,
 +
defined on a (topological) space  $  T $,
 +
viz.:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350222.png" /> are linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350223.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350224.png" />, called the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350225.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350226.png" />, or the coefficients of the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350227.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350228.png" />. Clearly, for the decomposition of an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350229.png" /> to exist, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350230.png" /> must be a complete set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350231.png" />, and for such a decomposition to be unique (i.e. for the zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350232.png" /> to have all components equal to zero), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350233.png" /> must be a topologically free set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350234.png" />.
+
$$
 +
\Xi (x): x  \in X  \rightarrow  \xi _ {x} (t)  \in K(T),
 +
$$
  
The sense and the practical significance of a topological basis (which will be simply denoted as a "basis" in what follows) is to establish a bijective linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350235.png" />, called the basis mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350236.png" /> into some (depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350237.png" />) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350238.png" /> of functions with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350239.png" />, defined on a (topological) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350240.png" />, viz.:
+
where  $  \xi _ {x} (t) = \xi _ {t} (x) $,
 +
so that, symbolically, $ \{ \xi _ {t} (X) \} = K(T) $
 +
and  $  \{ \xi _ {x} (T) \} = X $.  
 +
Owing to its concrete, effective definition, the structure of  $  K(T) $
 +
is simpler and more illustrative than that of the abstractly given  $  X $.  
 +
For instance, an algebraic basis of an infinite-dimensional Banach space is not countable, while in a number of cases, if the concept of a basis is suitably generalized, the cardinality of  $  T $
 +
is substantially smaller, and  $  K(T) $
 +
simplifies at the same time.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350241.png" /></td> </tr></table>
+
The space  $  K(T) $
 +
contains all functions of finite support, and the set of elements of the basis  $  \{ a _ {t} \} $
 +
is the bijective inverse image of the set of functions  $  \{ \xi _ {t} (s) \} $
 +
with only one non-zero value which is equal to one:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350242.png" />, so that, symbolically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350243.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350244.png" />. Owing to its concrete, effective definition, the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350245.png" /> is simpler and more illustrative than that of the abstractly given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350246.png" />. For instance, an algebraic basis of an infinite-dimensional Banach space is not countable, while in a number of cases, if the concept of a basis is suitably generalized, the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350247.png" /> is substantially smaller, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350248.png" /> simplifies at the same time.
+
$$
 +
a _ {t}  = \Xi  ^ {-1} [ \xi _ {t} (s) ],
 +
$$
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350249.png" /> contains all functions of finite support, and the set of elements of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350250.png" /> is the bijective inverse image of the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350251.png" /> with only one non-zero value which is equal to one:
+
where  $  \xi _ {t} (s) = 1 $
 +
if  $  t = s $,  
 +
and $  \xi _ {t} (s) = 0 $
 +
if  $  t \neq s $.  
 +
In other words,  $  a _ {t} $
 +
is the generator of a one-dimensional subspace  $  A _ {t} $
 +
which is complementary in  $  X $
 +
to the hyperplane defined by the equation  $  \xi _ {t} (x) = 0 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350252.png" /></td> </tr></table>
+
Thus, the role of the basis  $  \{ a _ {t} \} $
 +
is to organize, out of the set of components  $  \xi _ {t} (x) $
 +
which constitute the image of  $  x $
 +
under the basis mapping, a summable (in some sense) set  $  \{ \xi _ {t} (x) a _ {t} \} $,
 +
i.e. a basis  "decomposes" a space  $  X $
 +
into a (generalized) direct sum of one-dimensional subspaces:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350253.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350254.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350255.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350256.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350257.png" /> is the generator of a one-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350258.png" /> which is complementary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350259.png" /> to the hyperplane defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350260.png" />.
+
$$
 +
= \lim\limits  \sum \xi _ {t} (X)A _ {t} .
 +
$$
  
Thus, the role of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350261.png" /> is to organize, out of the set of components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350262.png" /> which constitute the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350263.png" /> under the basis mapping, a summable (in some sense) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350264.png" />, i.e. a basis  "decomposes"  a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350265.png" /> into a (generalized) direct sum of one-dimensional subspaces:
+
A basis is defined in a similar manner in vector spaces with a uniform, limit (pseudo-topological), linear ( $  L $-),  
 +
proximity, or other complementary structure.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350266.png" /></td> </tr></table>
+
Generalizations of the concept of a basis may be and in fact have been given in various directions. Thus, the introduction of a topology and a measure on  $  T $
 +
leads to the concept of the so-called continuous sum of elements from  $  X $
 +
and to corresponding integral representations; the decomposition of the space  $  X $
 +
into (not necessarily one-dimensional) components is used in the spectral theory of linear operators; the consideration of arbitrary topological algebras over a field  $  K $(
 +
e.g. algebras of measures on  $  T $
 +
with values in  $  K $
 +
or even in  $  X $,
 +
algebras of projection operators, etc.) instead of  $  K(T) $
 +
makes it possible to concretize many notions of abstract duality for topological vector spaces and, in particular, to employ the well-developed apparatus of the theory of characters.
  
A basis is defined in a similar manner in vector spaces with a uniform, limit (pseudo-topological), linear (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350267.png" />-), proximity, or other complementary structure.
+
A countable basis, which is the most extensively studied and, from the practical point of view, the most important example of a basis, is a sequence  $  \{ a _ {i} \} $
 +
of elements of a space  $  X $
 +
such that each element  $  x $
 +
is in unique correspondence with its series expansion with respect to the basis  $  \{ a _ {i} \} $
  
Generalizations of the concept of a basis may be and in fact have been given in various directions. Thus, the introduction of a topology and a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350268.png" /> leads to the concept of the so-called continuous sum of elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350269.png" /> and to corresponding integral representations; the decomposition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350270.png" /> into (not necessarily one-dimensional) components is used in the spectral theory of linear operators; the consideration of arbitrary topological algebras over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350271.png" /> (e.g. algebras of measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350272.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350273.png" /> or even in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350274.png" />, algebras of projection operators, etc.) instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350275.png" /> makes it possible to concretize many notions of abstract duality for topological vector spaces and, in particular, to employ the well-developed apparatus of the theory of characters.
+
$$
 +
\sum \xi _ {i} (x)a _ {i} ,\ \
 +
\xi _ {i} (x) \in K ,
 +
$$
  
A countable basis, which is the most extensively studied and, from the practical point of view, the most important example of a basis, is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350276.png" /> of elements of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350277.png" /> such that each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350278.png" /> is in unique correspondence with its series expansion with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350279.png" />
+
which (in the topology of  $  X $)
 +
converges to  $  x $.
 +
Here,  $  T = \mathbf Z $,
 +
and there exists a natural order in it. A countable basis is often simply called a  "basis" . A weak countable basis is defined in an analogous manner if weak convergence of the expansion is understood. For instance, the functions  $  e ^ {ikt } $,
 +
$  k \in \mathbf Z $,  
 +
form a basis in the spaces  $  L _ {p} $,
 +
$  1 < p < \infty $(
 +
periodic functions absolutely summable of degree  $  p $);
 +
on the contrary, these functions do not form a basis in the spaces  $  L _ {1} $,
 +
$  L _  \infty  $(
 +
measurable functions which almost everywhere coincide with bounded functions) or  $  C  ^ {1} $(
 +
continuous periodic functions). A necessary, but by far not sufficient, condition for the existence of a countable basis is the separability of  $  X $(
 +
e.g. a countable basis cannot exist in the space of measurable functions on an interval  $  [a, b] $
 +
with values in  $  \mathbf R $).  
 +
Moreover, the space  $  l _  \infty  $
 +
of bounded sequences, not being separable in the topology of $  l _  \infty  $,
 +
has no countable basis, but the elements a _ {i} = \{ \delta _ {ik }  \} $,
 +
where  $  \delta _ {ik }  = 1 $
 +
if  $  i=k $,
 +
and  $  \delta _ {ik }  = 0 $
 +
if  $  i \neq k $,
 +
form a basis in the weak topology  $  \sigma (l _  \infty  , l _ {1} ) $.  
 +
The question of the existence of a countable basis in separable Banach spaces (the basis problem) has been negatively solved [[#References|[6]]]. The analogous problem for nuclear spaces also has a negative solution [[#References|[7]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350280.png" /></td> </tr></table>
+
A countable basis is, however, not always  "well-suited" for applications. For example, the components  $  \xi _ {t} (x) $
 +
may be discontinuous, the expansion of  $  x $
 +
need not converge unconditionally, etc. In this connection one puts restrictions on the basis or introduces generalizations of it.
  
which (in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350281.png" />) converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350282.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350283.png" />, and there exists a natural order in it. A countable basis is often simply called "basis" . A weak countable basis is defined in an analogous manner if weak convergence of the expansion is understood. For instance, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350284.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350285.png" />, form a basis in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350286.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350287.png" /> (periodic functions absolutely summable of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350288.png" />); on the contrary, these functions do not form a basis in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350289.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350290.png" /> (measurable functions which almost everywhere coincide with bounded functions) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350291.png" /> (continuous periodic functions). A necessary, but by far not sufficient, condition for the existence of a countable basis is the separability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350292.png" /> (e.g. a countable basis cannot exist in the space of measurable functions on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350293.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350294.png" />). Moreover, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350295.png" /> of bounded sequences, not being separable in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350296.png" />, has no countable basis, but the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350297.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350298.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350299.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350300.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350301.png" />, form a basis in the weak topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350302.png" />. The question of the existence of a countable basis in separable Banach spaces (the basis problem) has been negatively solved [[#References|[6]]]. The analogous problem for nuclear spaces also has a negative solution [[#References|[7]]].
+
A basis of countable type is one of the generalizations of the concept of a countable basis in which, although  $  T $
 +
is not countable, nevertheless the decomposition of  $  x \in X $
 +
with respect to it has a natural definition: the corresponding space $  K(T) $
 +
consists of functions with countable support. For instance, a complete orthonormal set  $  \{ a _ {t} \} $
 +
in a Hilbert space  $  H $
 +
is a basis; if  $  x \in H $,  
 +
then  $  \xi _ {t} (x) = \langle x, a _ {t} \rangle $(
 +
where  $  \langle  \cdot , \cdot \rangle $
 +
is the scalar product in $  H $)  
 +
for all (except possibly a countable set of) indices  $  t \in T $,  
 +
and the series  $  \sum \xi _ {t} a _ {t} $
 +
converges to  $  x $.
 +
The basis mapping is determined by the orthogonal projections onto the closed subspaces generated by the elements  $  a _ {t} $.  
 +
A basis of the space $  AP $
 +
of all complex-valued almost-periodic functions on $  \mathbf R $
 +
consists of the functions  $  e ^ {i t \lambda } $;
 +
here, $  T = \mathbf R $,  
 +
$  K(T) $
 +
is the set of countably-valued functions, and the basis mapping is defined by the formula:
  
A countable basis is, however, not always "well-suited"  for applications. For example, the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350303.png" /> may be discontinuous, the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350304.png" /> need not converge unconditionally, etc. In this connection one puts restrictions on the basis or introduces generalizations of it.
+
$$
 +
\Xi [x( \lambda )] = \
 +
\lim\limits _ {\tau \rightarrow \infty } \
  
A basis of countable type is one of the generalizations of the concept of a countable basis in which, although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350305.png" /> is not countable, nevertheless the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350306.png" /> with respect to it has a natural definition: the corresponding space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350307.png" /> consists of functions with countable support. For instance, a complete orthonormal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350308.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350309.png" /> is a basis; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350310.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350311.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350312.png" /> is the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350313.png" />) for all (except possibly a countable set of) indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350314.png" />, and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350315.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350316.png" />. The basis mapping is determined by the orthogonal projections onto the closed subspaces generated by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350317.png" />. A basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350318.png" /> of all complex-valued almost-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350319.png" /> consists of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350320.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350321.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350322.png" /> is the set of countably-valued functions, and the basis mapping is defined by the formula:
+
\frac{1}{2 \tau }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350323.png" /></td> </tr></table>
+
\int\limits _ {- \tau } ^ { {+ }  \tau }
 +
x( \lambda )e ^ {it \lambda }  d \lambda .
 +
$$
  
An unconditional basis is a countable basis in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350324.png" /> such that the decomposition of any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350325.png" /> converges unconditionally (i.e. the sum of the series does not change if an arbitrary number of its terms is rearranged). For instance, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350326.png" /> (sequences converging to zero) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350327.png" /> (sequences summable of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350328.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350329.png" />) the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350330.png" /> form an unconditional basis; in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350331.png" /> of continuous functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350332.png" /> no basis can be unconditional. An orthonormal countable basis of a Hilbert space is an unconditional basis. A Banach space with an unconditional basis is weakly complete (accordingly, it has a separable dual space) if and only if it contains no subspace isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350333.png" /> (or, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350334.png" />).
+
An unconditional basis is a countable basis in a space $  X $
 +
such that the decomposition of any element $  x $
 +
converges unconditionally (i.e. the sum of the series does not change if an arbitrary number of its terms is rearranged). For instance, in $  c _ {0} $(
 +
sequences converging to zero) and $  l _ {p} $(
 +
sequences summable of degree $  p $,  
 +
$  1 \leq  p < \infty $)  
 +
the elements $  a _ {i} = \{ \delta _ {ik }  \} $
 +
form an unconditional basis; in the space $  C[a, b] $
 +
of continuous functions on the interval $  [a, b] $
 +
no basis can be unconditional. An orthonormal countable basis of a Hilbert space is an unconditional basis. A Banach space with an unconditional basis is weakly complete (accordingly, it has a separable dual space) if and only if it contains no subspace isomorphic to $  c _ {0} $(
 +
or, correspondingly, $  l _ {1} $).
  
Two bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350335.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350336.png" /> of the Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350337.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350338.png" />, respectively, are said to be equivalent if there exists a bijective linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350339.png" /> that can be extended to an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350340.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350341.png" />; these bases are said to be quasi-equivalent if they become equivalent as a result of a certain rearrangement and normalization of the elements of one of them. In each of the spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350342.png" /> all normalized unconditional bases are equivalent. However, there exist normalized bases not equivalent to orthonormal ones.
+
Two bases $  \{ a _ {i} \} $
 +
and $  \{ b _ {i} \} $
 +
of the Banach spaces $  X $
 +
and $  Y $,  
 +
respectively, are said to be equivalent if there exists a bijective linear mapping $  T : a _ {i} \rightarrow b _ {i} $
 +
that can be extended to an isomorphism between $  X $
 +
and $  Y $;  
 +
these bases are said to be quasi-equivalent if they become equivalent as a result of a certain rearrangement and normalization of the elements of one of them. In each of the spaces, $  l _ {1} , l _ {2} , c _ {0} $
 +
all normalized unconditional bases are equivalent. However, there exist normalized bases not equivalent to orthonormal ones.
  
A summable basis — a generalization of the concept of an unconditional basis corresponding to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350343.png" /> of arbitrary cardinality and becoming identical with it if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350344.png" /> — is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350345.png" /> such that for an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350346.png" /> there exists a set of linear combinations (partial sums) of elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350347.png" />, which is called a generalized decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350348.png" />, which is summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350349.png" />. This means that for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350350.png" /> of zero it is possible to find a finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350351.png" /> such that for any finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350352.png" /> the relation
+
A summable basis — a generalization of the concept of an unconditional basis corresponding to a set $  T $
 +
of arbitrary cardinality and becoming identical with it if $  T = \mathbf Z $—  
 +
is a set $  A = \{ {a _ {t} } : {t \in T } \} $
 +
such that for an arbitrary element $  x \in X $
 +
there exists a set of linear combinations (partial sums) of elements from $  A $,  
 +
which is called a generalized decomposition of $  x $,  
 +
which is summable to $  x $.  
 +
This means that for any neighbourhood $  U \subset  X $
 +
of zero it is possible to find a finite subset $  A _ {U} \subset  A $
 +
such that for any finite set $  A  ^  \prime  \supset A _ {U} $
 +
the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350353.png" /></td> </tr></table>
+
$$
 +
\left ( \sum _ {t \in A  ^  \prime  }
 +
\xi _ {t} a _ {t} - x \right )  \in  U,
 +
$$
  
is true, i.e. when the partial sums form a Cauchy system (Cauchy filter). For instance, an arbitrary orthonormal basis of a Hilbert space is a summable basis. A weakly summable basis is defined in a similar way. A totally summable basis is a summable basis such that there exists a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350354.png" /> for which the set of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350355.png" /> is summable. A totally summable basis is at most countable. In a dual nuclear space all weakly summable bases are totally summable.
+
is true, i.e. when the partial sums form a Cauchy system (Cauchy filter). For instance, an arbitrary orthonormal basis of a Hilbert space is a summable basis. A weakly summable basis is defined in a similar way. A totally summable basis is a summable basis such that there exists a bounded set $  B $
 +
for which the set of semi-norms $  \{ p _ {B} ( \xi _ {t} a _ {t} ) \} $
 +
is summable. A totally summable basis is at most countable. In a dual nuclear space all weakly summable bases are totally summable.
  
An absolute basis (absolutely summable basis) is a summable basis of a locally convex space over a normed field such that for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350356.png" /> of zero and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350357.png" /> the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350358.png" /> is summable. All unconditional countable bases are absolute, i.e. the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350359.png" /> converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350360.png" /> and all continuous semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350361.png" />. Of all Banach spaces only the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350362.png" /> has an absolute countable basis. If a Fréchet space has an absolute basis, all its unconditional bases are absolute. In nuclear Fréchet spaces any countable basis (if it exists) is absolute [[#References|[13]]].
+
An absolute basis (absolutely summable basis) is a summable basis of a locally convex space over a normed field such that for any neighbourhood $  U $
 +
of zero and for each $  t \in T $
 +
the family of semi-norms $  \{ p _ {U} (a _ {t} ) \} $
 +
is summable. All unconditional countable bases are absolute, i.e. the series $  \sum | \xi _ {i} (x) |  p ( a _ {i} ) $
 +
converges for all $  x \in X $
 +
and all continuous semi-norms $  p ( \cdot ) $.  
 +
Of all Banach spaces only the space $  l _ {1} $
 +
has an absolute countable basis. If a Fréchet space has an absolute basis, all its unconditional bases are absolute. In nuclear Fréchet spaces any countable basis (if it exists) is absolute [[#References|[13]]].
  
A Schauder basis is a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350363.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350364.png" /> such that the basis mapping defined by it is continuous (and is therefore an isomorphism onto some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350365.png" />), i.e. a basis in which the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350366.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350367.png" /> and, in particular, the coefficients of the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350368.png" /> with respect to this basis, are continuous functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350369.png" />. This basis was first defined by J. Schauder [[#References|[5]]] for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350370.png" />. The concept of a Schauder basis is the most important of all modifications of the concept of a basis.
+
A Schauder basis is a basis $  \{ {a _ {t} } : {t \in T } \} $
 +
of a space $  X $
 +
such that the basis mapping defined by it is continuous (and is therefore an isomorphism onto some space $  K(T) $),  
 +
i.e. a basis in which the components $  \xi _ {t} (x) $
 +
for any $  x \in X $
 +
and, in particular, the coefficients of the decomposition of $  x $
 +
with respect to this basis, are continuous functionals on $  X $.  
 +
This basis was first defined by J. Schauder [[#References|[5]]] for the case $  T = \mathbf Z $.  
 +
The concept of a Schauder basis is the most important of all modifications of the concept of a basis.
  
A Schauder basis is characterized by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350371.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350372.png" /> form a bi-orthogonal system. Thus, the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350373.png" /> form countable Schauder bases in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350374.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350375.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350376.png" />. A countable Schauder basis forms a [[Haar system|Haar system]] in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350377.png" />. In complete metric vector spaces (in particular, in Banach spaces) all countable bases are Schauder bases [[#References|[10]]]. In Fréchet spaces the concept of a weak basis and a Schauder basis are identical [[#References|[11]]]. In barrelled spaces in which there are no linear continuous functionals, there is also no Schauder basis [[#References|[8]]]. However, if a weak Schauder basis exists in these spaces, it is an ordinary Schauder basis [[#References|[9]]]. A barrelled locally convex space with a countable Schauder basis is reflexive if and only if this basis is at the same time a shrinking set, i.e. if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350378.png" /> corresponding to it will be a basis in the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350379.png" /> and will be boundedly complete, i.e. if the boundedness of the set of partial sums of a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350380.png" /> implies that this series is convergent [[#References|[12]]]. If a Schauder basis is an unconditional basis in a Banach space, then it is a shrinking set (or a boundedly complete set) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350381.png" /> does not contain subspaces isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350382.png" /> (or, respectively, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350383.png" />).
+
A Schauder basis is characterized by the fact that $  \{ a _ {t} \} $
 +
and $  \{ \xi _ {t} \} $
 +
form a [[biorthogonal system]]. Thus, the sequences $  a _ {i} = \{ \delta _ {ik }  \} $
 +
form countable Schauder bases in the spaces $  c _ {0} $
 +
and $  l _ {p} $,  
 +
$  p \geq  1 $.  
 +
A countable Schauder basis forms a [[Haar system|Haar system]] in the space $  C[a, b] $.  
 +
In complete metric vector spaces (in particular, in Banach spaces) all countable bases are Schauder bases [[#References|[10]]]. In Fréchet spaces the concept of a weak basis and a Schauder basis are identical [[#References|[11]]]. In barrelled spaces in which there are no linear continuous functionals, there is also no Schauder basis [[#References|[8]]]. However, if a weak Schauder basis exists in these spaces, it is an ordinary Schauder basis [[#References|[9]]]. A barrelled locally convex space with a countable Schauder basis is reflexive if and only if this basis is at the same time a shrinking set, i.e. if the $  \{ \xi _ {t} \} $
 +
corresponding to it will be a basis in the dual space $  X  ^ {*} $
 +
and will be boundedly complete, i.e. if the boundedness of the set of partial sums of a series $  \sum _ {i} \xi _ {i} a _ {i} $
 +
implies that this series is convergent [[#References|[12]]]. If a Schauder basis is an unconditional basis in a Banach space, then it is a shrinking set (or a boundedly complete set) if and only if $  X $
 +
does not contain subspaces isomorphic to $  l _ {1} $(
 +
or, respectively, to $  c _ {0} $).
  
A Schauder basis in a locally convex space is equicontinuous if for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350384.png" /> of zero it is possible to find a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350385.png" /> of zero such that
+
A Schauder basis in a locally convex space is equicontinuous if for any neighbourhood $  U $
 +
of zero it is possible to find a neighbourhood $  V $
 +
of zero such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350386.png" /></td> </tr></table>
+
$$
 +
| \xi _ {t} (x) | \
 +
p _ {U} (a _ {t} )  \leq  p _ {V} ( x )
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350387.png" />. All Schauder bases of a barrelled space are equicontinuous, and each complete locally convex space with a countable equicontinuous basis can be identified with some sequence space [[#References|[15]]]. An equicontinuous basis of a nuclear space is absolute.
+
for all $  x \in X, t \in T $.  
 +
All Schauder bases of a barrelled space are equicontinuous, and each complete locally convex space with a countable equicontinuous basis can be identified with some sequence space [[#References|[15]]]. An equicontinuous basis of a nuclear space is absolute.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Hamel,  "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015350/b015350388.png" />"  ''Math. Ann.'' , '''60'''  (1905)  pp. 459–462</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Schauder,  "Zur Theorie stetiger Abbildungen in Funktionalräumen"  ''Math. Z.'' , '''26'''  (1927)  pp. 47–65; 417–431</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Enflo,  "A counterexample to the approximation problem in Banach spaces"  ''Acta Math.'' , '''130'''  (1973)  pp. 309–317</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.M. Zobin,  B.S. Mityagin,  "Examples of nuclear linear metric spaces without a basis"  ''Functional Anal. Appl.'' , '''8''' :  4  (1974)  pp. 304–313  ''Funktsional. Analiz. i Prilozhen.'' , '''8''' :  4  (1974)  pp. 35–47</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J. Dieudonné,  "Sur les espaces de Köthe"  ''J. d'Anal. Math.'' , '''1'''  (1951)  pp. 81–115</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.G. Arsove,  "The Paley-Wiener theorem in metric linear spaces"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 365–379</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  C. Bessaga,  A. Pelczyński,  "Spaces of continuous functions IV"  ''Studia Math.'' , '''19'''  (1960)  pp. 53–62</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.C. James,  "Bases and reflexivity in Banach spaces"  ''Ann. of Math. (2)'' , '''52''' :  3  (1950)  pp. 518–527</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A. Dynin,  B. Mityagin,  "Criterion for nuclearity in terms of approximate dimension"  ''Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys.'' , '''8'''  (1960)  pp. 535–540</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  I.M. Singer,  "Bases in Banach spaces" , '''1–2''' , Springer  (1970–1981)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  G. Hamel,  "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x+y)=f(x)+f(y)$"  ''Math. Ann.'' , '''60'''  (1905)  pp. 459–462</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Schauder,  "Zur Theorie stetiger Abbildungen in Funktionalräumen"  ''Math. Z.'' , '''26'''  (1927)  pp. 47–65; 417–431</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Enflo,  "A counterexample to the approximation problem in Banach spaces"  ''Acta Math.'' , '''130'''  (1973)  pp. 309–317</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.M. Zobin,  B.S. Mityagin,  "Examples of nuclear linear metric spaces without a basis"  ''Functional Anal. Appl.'' , '''8''' :  4  (1974)  pp. 304–313  ''Funktsional. Analiz. i Prilozhen.'' , '''8''' :  4  (1974)  pp. 35–47</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J. Dieudonné,  "Sur les espaces de Köthe"  ''J. d'Anal. Math.'' , '''1'''  (1951)  pp. 81–115</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.G. Arsove,  "The Paley-Wiener theorem in metric linear spaces"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 365–379</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  C. Bessaga,  A. Pelczyński,  "Spaces of continuous functions IV"  ''Studia Math.'' , '''19'''  (1960)  pp. 53–62</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.C. James,  "Bases and reflexivity in Banach spaces"  ''Ann. of Math. (2)'' , '''52''' :  3  (1950)  pp. 518–527</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A. Dynin,  B. Mityagin,  "Criterion for nuclearity in terms of approximate dimension"  ''Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys.'' , '''8'''  (1960)  pp. 535–540</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  I.M. Singer,  "Bases in Banach spaces" , '''1–2''' , Springer  (1970–1981)</TD></TR>
 +
</table>

Latest revision as of 07:37, 26 March 2023


of a set $ X $

A minimal subset $ B $ that generates it. Generation here means that by application of operations of a certain class $ \Omega $ to elements $ b \in B $ it is possible to obtain any element $ x \in X $. This concept is related to the concept of dependence: By means of operations from $ \Omega $ the elements of $ X $ become dependent on the elements of $ B $. Minimality means that no proper subset $ B _ {1} \subset B $ generates $ X $. In a certain sense this property causes the elements of $ B $ to be independent: None of the elements $ b \in B $ is generated by the other elements of $ B $. For instance, the set of all natural numbers $ \mathbf Z _ {0} $ has the unique element 0 as basis and is generated from it by the operation of immediate succession and its iteration. The set of all natural numbers $ >1 $ is generated by the operation of multiplication from the basis consisting of all prime numbers. A basis of the algebra of quaternions consists of the four elements $ \{ 1, i, j, k \} $ if the generating operations consist of addition and of multiplication by real numbers; if, in addition to these operations, one also includes multiplication of quaternions, the basis will consist of three elements only — $ \{ 1, i, j \} $( because $ k=ij $).

A basis of the natural numbers of order $ k $ is a subsequence $ \Omega $ of the set $ \mathbf Z _ {0} $ of natural numbers including 0, which, as a result of $ k $- fold addition to itself (the generating operation) yields all of $ \mathbf Z _ {0} $. This means that any natural number $ n $ can be represented in the form

$$ n = a _ {1} + \dots + a _ {k} , $$

where $ a _ {i} \in \Omega $. For example, every natural number is a sum of four squares of natural numbers (Lagrange's theorem), i.e. the sequence of squares is a basis of $ \mathbf Z _ {0} $ of order 4. In general, the sequence of $ m $- th powers of natural numbers is a basis of $ \mathbf Z _ {0} $( Hilbert's theorem), the order of which has been estimated by the Vinogradov method. The concept of a basis of $ \mathbf Z _ {0} $ has been generalized to the case of arbitrary sequences of numbers, i.e. functions on $ \mathbf Z _ {0} $.

A set $ X $ always contains a generating set (in the trivial case: $ X $ generates $ X $), but minimality may prove to be principally impossible (such a situation is typical of classes $ \Omega $ containing infinite-place operations, in particular in topological structures, lattices, etc.). For this reason the minimality condition is replaced by a weaker requirement: A basis is a generating set of minimal cardinality. In this context a basis $ B $ is defined as a parametrized set (or population), i.e. as a function $ b(t) $ on a set of indices $ T $ with values in $ X $, such that $ b(T) = B $; the cardinality of $ T $ is sometimes called as the dimension (or rank) of the basis of $ X $. For example, a countable everywhere-dense set $ B $ in a separable topological space $ P $ may be considered as a basis for it; $ P $ is generated from $ B $ by the closure operation (which, incidentally, is related to generation in more general cases as well, see below).

A basis for a topology of a topological space $ X $( a base) is a basis $ \mathfrak B $ of the set of all open subsets in $ X $; the generation is effected by taking unions of elements of $ \mathfrak B $.

A basis of a Boolean algebra $ \mathfrak A $( a dual base of $ \mathfrak A $ in the sense of Tarski) is a dense set $ S $( of minimal cardinality) in $ \mathfrak A $; the generation of $ \mathfrak A $ from $ S $( and hence $ S $ itself) is determined by the condition $ s \rightarrow a = \lor $( which is equivalent to $ s \subset a $), where $ s \in S $, $ a \in \mathfrak A $, $ \lor $ is the unit of $ \mathfrak A $ and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter $ \nabla $ as a set $ S $ such that for an arbitrary $ a \in \nabla $ there exists an $ s \in S $ with $ s \subset a $.

More special cases of bases of a set $ X $ are introduced according to the following procedure. Let $ B(X) $ be the Boolean algebra of $ X $, i.e. the set of all its subsets. A generating operator (or a closure operator) $ J $ is a mapping of $ B (X) $ into itself such that if $ A \subset B $, then $ J(A) \subset J(B) $; $ A \subset J(A) $; $ JJ(A) = J(A) $.

An element $ x \in X $ is generated by a set $ A $ if $ x \in J(A) $; in particular, $ A $ generates $ X $ if $ J(A) = X $. A minimal set $ B $ possessing this property is said to be a basis of $ X $ defined by the operator $ J $. A generating operator $ J $ is of finite type if, for arbitrary $ A \subset X $ and $ x \subset X $, it follows from $ x \in J(A) $ that $ x \in J(A _ {0} ) $ for a certain finite subset $ A _ {0} \subset A $; a generating operator $ J $ has the property of substitution if, for any $ y, z \in X $ and $ A \subset X $, both $ y \notin J(A) $ and $ y \in J(A \cup \{ z \} ) $ imply that $ z \in J(A \cup \{ y \} ) $. A generating operator $ J $ of finite type with the substitution property defines a dependence relation on $ X $, i.e. a subdivision of $ B(X) $ into two classes — dependent and independent sets; a set $ A $ is said to be dependent if $ y \in J(A \setminus y) $ for some $ y \in A $, and is said to be independent if $ y \notin J (A \setminus y) $ for any $ y \in A $. Therefore, $ A $ is dependent (independent) if and only if some (arbitrary) non-empty finite subset(s) $ A _ {0} \subset A $ is dependent (are independent).

For a set $ B $ to be a basis of the set $ X $ it is necessary and sufficient for $ B $ to be an independent generating set for $ X $, or else, a maximal independent set in $ X $.

If $ A $ is an arbitrary independent set, and $ C $ is an $ X $- generating set containing $ A $, then there exists a basis $ B $ in $ X $ such that $ A \subset B \subset C $. In particular, $ X $ always has a basis, and any two bases of it have the same cardinality.

In algebraic systems $ X $ an important role is played by the concept of the so-called free basis $ B $, which is characterized by the following property: Any mapping of $ B \subset X $ into any algebraic system $ Y $( of the same signature) may be extended to a (unique) (homo)morphism from $ X $ into $ Y $ or, which is the same thing, for any (homo)morphism $ \theta : X \rightarrow Y $ and any set $ A \subset X $, the generating operators $ J _ {X} $ and $ J _ {Y} $ satisfy the condition:

$$ \theta \{ J _ {X} (A) \} = \ J _ {Y} ( \theta \{ A \} ) . $$

An algebraic system with a free basis is said to be free.

A typical example is a basis of a (unitary) module $ M $ over a ring $ K $, that is, a free family of elements from $ M $ generating $ M $[3]. Here, a family $ A = \{ {a _ {t} } : {t \in T } \} $ of elements of a $ K $- module $ M $ is said to be free if $ \sum \xi _ {t} a _ {t} = 0 $( where $ \xi _ {t} = 0 $ for all except a finite number of indices $ t $) implies that $ \xi _ {t} = 0 $ for all $ t $, and the generation is realized by representing the elements $ x $ as linear combinations of the elements $ a _ {t} $: There exists a set (dependent on $ x $) of elements $ \xi _ {t} \in K $ such that $ \xi _ {t} = 0 $ for all except a finite number of indices $ t $, and such that the decomposition

$$ x = \sum \xi _ {t} a _ {t} $$

is valid (i.e. $ X $ is the linear envelope of $ A $). In this sense, the basis $ M $ is free basis; the converse proposition is also true. Thus, the set of periods of a doubly-periodic function $ f $ of one complex variable, which is a discrete Abelian group (and hence a module over the ring $ \mathbf Z $), has a free basis, called the period basis of $ f $; it consists of two so-called primitive periods. A period basis of an Abelian function of several complex variables is defined in a similar manner.

If $ K $ is a skew-field, all bases (in the previous sense) are free. On the contrary, there exist modules without a free basis; these include, for example, the non-principal ideals in an integral domain $ K $, considered as a $ K $- module.

A basis of a vector space $ X $ over a field $ K $ is a (free) basis of the unitary module which underlies $ X $. In a similar manner, a basis of an algebra $ A $ over a field $ K $ is a basis of the vector space underlying $ A $. All bases of a given vector space $ X $ have the same cardinality, which is equal to the cardinality of $ T $; the latter is called the algebraic dimension of $ X $. Each element $ x \in X $ can be represented as a linear combination of basis elements in a unique way. The elements $ \xi _ {t} (x) \in K $, which are linear functionals on $ X $, are called the components (coordinates) of $ x $ in the given basis $ \{ a _ {t} \} $.

A set $ A $ is a basis in $ X $ if and only if $ A $ is a maximal (with respect to inclusion) free set in $ X $.

The mapping

$$ \Xi : x \rightarrow \xi _ {x} (t), $$

where $ \xi _ {x} (t) = \xi _ {t} (x) $ if $ \xi _ {t} $ is the value of the $ t $- th component of $ x $ in the basis $ A $, and 0 otherwise, is called the basis mapping; it is a linear injective mapping of $ X $ into the space $ K ^ {T} $ of functions on $ T $ with values in $ K $. In this case the image $ \Xi (X) $ consists of all functions with a finite number of non-zero values (functions of finite support). This interpretation permits one to define a generalized basis of a vector space $ X $ over a field $ K $ as a bijective linear mapping from it to some subspace $ K (T) $ of the space $ K ^ {T} $ of functions on $ T $ with values in $ K $, where $ T $ is some suitably chosen set. However, unless additional restrictions (e.g. an order) and additional structures (e.g. a topology) are imposed on $ T $, and corresponding compatible conditions on $ K(T) $ are introduced, the concept of a generalized basis is seldom of use in practice.

A basis of a vector space $ X $ is sometimes called an algebraic basis; in this way it is stressed that there is no connection with additional structures on $ X $, even if they are compatible with its vector structure.

A Hamel basis is a basis of the field of real numbers $ \mathbf R $, considered as a vector space over the field of rational numbers. It was introduced by G. Hamel [4] to obtain a discontinuous solution of the functional equation $ f(x+y) = f(x) + f(y) $; the graph of its solution is everywhere dense in the plane $ \mathbf R ^ {2} $. To each almost-periodic function corresponds some countable Hamel basis $ \beta $ such that each Fourier exponent $ \Lambda _ {n} $ of this function belongs to the linear envelope of $ \beta $. The elements of $ \beta $ may be so chosen that they belong to a sequence $ \{ \Lambda _ {i} \} $; the set $ \beta $ is said to be a basis of the almost-periodic functions. An analogous basis has been constructed in a ring containing a skew-field $ P $ and which has the unit of $ P $ as its own unit. An algebraic basis of an arbitrary vector space is also sometimes referred to as a Hamel basis.

A topological basis (a basis of a topological vector space $ X $ over a field $ K $) is a set $ A = \{ {a _ {t} } : {t \in T } \} \subset X $ with properties and functions analogous to those of the algebraic basis of the vector space. The concept of a topological basis, which is one of the most important ones in functional analysis, generalizes the concept of an algebraic basis with regard to the topological structure of $ X $ and makes it possible to obtain, for each element $ X $, its decomposition with respect to the basis $ \{ a _ {t} \} $, which is moreover unique, i.e. a representation of $ x $ as a limit (in some sense) of linear combinations of elements $ a _ {t} $:

$$ x = \lim\limits \sum \xi _ {t} (x)a _ {t} , $$

where $ \xi _ {t} (x) $ are linear functionals on $ X $ with values in $ K $, called the components of $ x $ in the basis $ A $, or the coefficients of the decomposition of $ x $ with respect to the basis $ A $. Clearly, for the decomposition of an arbitrary $ x $ to exist, $ A $ must be a complete set in $ X $, and for such a decomposition to be unique (i.e. for the zero element of $ X $ to have all components equal to zero), $ A $ must be a topologically free set in $ X $.

The sense and the practical significance of a topological basis (which will be simply denoted as a "basis" in what follows) is to establish a bijective linear mapping of $ X $, called the basis mapping, $ \Xi $ into some (depending on $ X $) space $ K(T) $ of functions with values in $ K $, defined on a (topological) space $ T $, viz.:

$$ \Xi (x): x \in X \rightarrow \xi _ {x} (t) \in K(T), $$

where $ \xi _ {x} (t) = \xi _ {t} (x) $, so that, symbolically, $ \{ \xi _ {t} (X) \} = K(T) $ and $ \{ \xi _ {x} (T) \} = X $. Owing to its concrete, effective definition, the structure of $ K(T) $ is simpler and more illustrative than that of the abstractly given $ X $. For instance, an algebraic basis of an infinite-dimensional Banach space is not countable, while in a number of cases, if the concept of a basis is suitably generalized, the cardinality of $ T $ is substantially smaller, and $ K(T) $ simplifies at the same time.

The space $ K(T) $ contains all functions of finite support, and the set of elements of the basis $ \{ a _ {t} \} $ is the bijective inverse image of the set of functions $ \{ \xi _ {t} (s) \} $ with only one non-zero value which is equal to one:

$$ a _ {t} = \Xi ^ {-1} [ \xi _ {t} (s) ], $$

where $ \xi _ {t} (s) = 1 $ if $ t = s $, and $ \xi _ {t} (s) = 0 $ if $ t \neq s $. In other words, $ a _ {t} $ is the generator of a one-dimensional subspace $ A _ {t} $ which is complementary in $ X $ to the hyperplane defined by the equation $ \xi _ {t} (x) = 0 $.

Thus, the role of the basis $ \{ a _ {t} \} $ is to organize, out of the set of components $ \xi _ {t} (x) $ which constitute the image of $ x $ under the basis mapping, a summable (in some sense) set $ \{ \xi _ {t} (x) a _ {t} \} $, i.e. a basis "decomposes" a space $ X $ into a (generalized) direct sum of one-dimensional subspaces:

$$ X = \lim\limits \sum \xi _ {t} (X)A _ {t} . $$

A basis is defined in a similar manner in vector spaces with a uniform, limit (pseudo-topological), linear ( $ L $-), proximity, or other complementary structure.

Generalizations of the concept of a basis may be and in fact have been given in various directions. Thus, the introduction of a topology and a measure on $ T $ leads to the concept of the so-called continuous sum of elements from $ X $ and to corresponding integral representations; the decomposition of the space $ X $ into (not necessarily one-dimensional) components is used in the spectral theory of linear operators; the consideration of arbitrary topological algebras over a field $ K $( e.g. algebras of measures on $ T $ with values in $ K $ or even in $ X $, algebras of projection operators, etc.) instead of $ K(T) $ makes it possible to concretize many notions of abstract duality for topological vector spaces and, in particular, to employ the well-developed apparatus of the theory of characters.

A countable basis, which is the most extensively studied and, from the practical point of view, the most important example of a basis, is a sequence $ \{ a _ {i} \} $ of elements of a space $ X $ such that each element $ x $ is in unique correspondence with its series expansion with respect to the basis $ \{ a _ {i} \} $

$$ \sum \xi _ {i} (x)a _ {i} ,\ \ \xi _ {i} (x) \in K , $$

which (in the topology of $ X $) converges to $ x $. Here, $ T = \mathbf Z $, and there exists a natural order in it. A countable basis is often simply called a "basis" . A weak countable basis is defined in an analogous manner if weak convergence of the expansion is understood. For instance, the functions $ e ^ {ikt } $, $ k \in \mathbf Z $, form a basis in the spaces $ L _ {p} $, $ 1 < p < \infty $( periodic functions absolutely summable of degree $ p $); on the contrary, these functions do not form a basis in the spaces $ L _ {1} $, $ L _ \infty $( measurable functions which almost everywhere coincide with bounded functions) or $ C ^ {1} $( continuous periodic functions). A necessary, but by far not sufficient, condition for the existence of a countable basis is the separability of $ X $( e.g. a countable basis cannot exist in the space of measurable functions on an interval $ [a, b] $ with values in $ \mathbf R $). Moreover, the space $ l _ \infty $ of bounded sequences, not being separable in the topology of $ l _ \infty $, has no countable basis, but the elements $ a _ {i} = \{ \delta _ {ik } \} $, where $ \delta _ {ik } = 1 $ if $ i=k $, and $ \delta _ {ik } = 0 $ if $ i \neq k $, form a basis in the weak topology $ \sigma (l _ \infty , l _ {1} ) $. The question of the existence of a countable basis in separable Banach spaces (the basis problem) has been negatively solved [6]. The analogous problem for nuclear spaces also has a negative solution [7].

A countable basis is, however, not always "well-suited" for applications. For example, the components $ \xi _ {t} (x) $ may be discontinuous, the expansion of $ x $ need not converge unconditionally, etc. In this connection one puts restrictions on the basis or introduces generalizations of it.

A basis of countable type is one of the generalizations of the concept of a countable basis in which, although $ T $ is not countable, nevertheless the decomposition of $ x \in X $ with respect to it has a natural definition: the corresponding space $ K(T) $ consists of functions with countable support. For instance, a complete orthonormal set $ \{ a _ {t} \} $ in a Hilbert space $ H $ is a basis; if $ x \in H $, then $ \xi _ {t} (x) = \langle x, a _ {t} \rangle $( where $ \langle \cdot , \cdot \rangle $ is the scalar product in $ H $) for all (except possibly a countable set of) indices $ t \in T $, and the series $ \sum \xi _ {t} a _ {t} $ converges to $ x $. The basis mapping is determined by the orthogonal projections onto the closed subspaces generated by the elements $ a _ {t} $. A basis of the space $ AP $ of all complex-valued almost-periodic functions on $ \mathbf R $ consists of the functions $ e ^ {i t \lambda } $; here, $ T = \mathbf R $, $ K(T) $ is the set of countably-valued functions, and the basis mapping is defined by the formula:

$$ \Xi [x( \lambda )] = \ \lim\limits _ {\tau \rightarrow \infty } \ \frac{1}{2 \tau } \int\limits _ {- \tau } ^ { {+ } \tau } x( \lambda )e ^ {it \lambda } d \lambda . $$

An unconditional basis is a countable basis in a space $ X $ such that the decomposition of any element $ x $ converges unconditionally (i.e. the sum of the series does not change if an arbitrary number of its terms is rearranged). For instance, in $ c _ {0} $( sequences converging to zero) and $ l _ {p} $( sequences summable of degree $ p $, $ 1 \leq p < \infty $) the elements $ a _ {i} = \{ \delta _ {ik } \} $ form an unconditional basis; in the space $ C[a, b] $ of continuous functions on the interval $ [a, b] $ no basis can be unconditional. An orthonormal countable basis of a Hilbert space is an unconditional basis. A Banach space with an unconditional basis is weakly complete (accordingly, it has a separable dual space) if and only if it contains no subspace isomorphic to $ c _ {0} $( or, correspondingly, $ l _ {1} $).

Two bases $ \{ a _ {i} \} $ and $ \{ b _ {i} \} $ of the Banach spaces $ X $ and $ Y $, respectively, are said to be equivalent if there exists a bijective linear mapping $ T : a _ {i} \rightarrow b _ {i} $ that can be extended to an isomorphism between $ X $ and $ Y $; these bases are said to be quasi-equivalent if they become equivalent as a result of a certain rearrangement and normalization of the elements of one of them. In each of the spaces, $ l _ {1} , l _ {2} , c _ {0} $ all normalized unconditional bases are equivalent. However, there exist normalized bases not equivalent to orthonormal ones.

A summable basis — a generalization of the concept of an unconditional basis corresponding to a set $ T $ of arbitrary cardinality and becoming identical with it if $ T = \mathbf Z $— is a set $ A = \{ {a _ {t} } : {t \in T } \} $ such that for an arbitrary element $ x \in X $ there exists a set of linear combinations (partial sums) of elements from $ A $, which is called a generalized decomposition of $ x $, which is summable to $ x $. This means that for any neighbourhood $ U \subset X $ of zero it is possible to find a finite subset $ A _ {U} \subset A $ such that for any finite set $ A ^ \prime \supset A _ {U} $ the relation

$$ \left ( \sum _ {t \in A ^ \prime } \xi _ {t} a _ {t} - x \right ) \in U, $$

is true, i.e. when the partial sums form a Cauchy system (Cauchy filter). For instance, an arbitrary orthonormal basis of a Hilbert space is a summable basis. A weakly summable basis is defined in a similar way. A totally summable basis is a summable basis such that there exists a bounded set $ B $ for which the set of semi-norms $ \{ p _ {B} ( \xi _ {t} a _ {t} ) \} $ is summable. A totally summable basis is at most countable. In a dual nuclear space all weakly summable bases are totally summable.

An absolute basis (absolutely summable basis) is a summable basis of a locally convex space over a normed field such that for any neighbourhood $ U $ of zero and for each $ t \in T $ the family of semi-norms $ \{ p _ {U} (a _ {t} ) \} $ is summable. All unconditional countable bases are absolute, i.e. the series $ \sum | \xi _ {i} (x) | p ( a _ {i} ) $ converges for all $ x \in X $ and all continuous semi-norms $ p ( \cdot ) $. Of all Banach spaces only the space $ l _ {1} $ has an absolute countable basis. If a Fréchet space has an absolute basis, all its unconditional bases are absolute. In nuclear Fréchet spaces any countable basis (if it exists) is absolute [13].

A Schauder basis is a basis $ \{ {a _ {t} } : {t \in T } \} $ of a space $ X $ such that the basis mapping defined by it is continuous (and is therefore an isomorphism onto some space $ K(T) $), i.e. a basis in which the components $ \xi _ {t} (x) $ for any $ x \in X $ and, in particular, the coefficients of the decomposition of $ x $ with respect to this basis, are continuous functionals on $ X $. This basis was first defined by J. Schauder [5] for the case $ T = \mathbf Z $. The concept of a Schauder basis is the most important of all modifications of the concept of a basis.

A Schauder basis is characterized by the fact that $ \{ a _ {t} \} $ and $ \{ \xi _ {t} \} $ form a biorthogonal system. Thus, the sequences $ a _ {i} = \{ \delta _ {ik } \} $ form countable Schauder bases in the spaces $ c _ {0} $ and $ l _ {p} $, $ p \geq 1 $. A countable Schauder basis forms a Haar system in the space $ C[a, b] $. In complete metric vector spaces (in particular, in Banach spaces) all countable bases are Schauder bases [10]. In Fréchet spaces the concept of a weak basis and a Schauder basis are identical [11]. In barrelled spaces in which there are no linear continuous functionals, there is also no Schauder basis [8]. However, if a weak Schauder basis exists in these spaces, it is an ordinary Schauder basis [9]. A barrelled locally convex space with a countable Schauder basis is reflexive if and only if this basis is at the same time a shrinking set, i.e. if the $ \{ \xi _ {t} \} $ corresponding to it will be a basis in the dual space $ X ^ {*} $ and will be boundedly complete, i.e. if the boundedness of the set of partial sums of a series $ \sum _ {i} \xi _ {i} a _ {i} $ implies that this series is convergent [12]. If a Schauder basis is an unconditional basis in a Banach space, then it is a shrinking set (or a boundedly complete set) if and only if $ X $ does not contain subspaces isomorphic to $ l _ {1} $( or, respectively, to $ c _ {0} $).

A Schauder basis in a locally convex space is equicontinuous if for any neighbourhood $ U $ of zero it is possible to find a neighbourhood $ V $ of zero such that

$$ | \xi _ {t} (x) | \ p _ {U} (a _ {t} ) \leq p _ {V} ( x ) $$

for all $ x \in X, t \in T $. All Schauder bases of a barrelled space are equicontinuous, and each complete locally convex space with a countable equicontinuous basis can be identified with some sequence space [15]. An equicontinuous basis of a nuclear space is absolute.

References

[1] P.M. Cohn, "Universal algebra" , Reidel (1981)
[2] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[3] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[4] G. Hamel, "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x+y)=f(x)+f(y)$" Math. Ann. , 60 (1905) pp. 459–462
[5] J. Schauder, "Zur Theorie stetiger Abbildungen in Funktionalräumen" Math. Z. , 26 (1927) pp. 47–65; 417–431
[6] P. Enflo, "A counterexample to the approximation problem in Banach spaces" Acta Math. , 130 (1973) pp. 309–317
[7] N.M. Zobin, B.S. Mityagin, "Examples of nuclear linear metric spaces without a basis" Functional Anal. Appl. , 8 : 4 (1974) pp. 304–313 Funktsional. Analiz. i Prilozhen. , 8 : 4 (1974) pp. 35–47
[8] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
[9] J. Dieudonné, "Sur les espaces de Köthe" J. d'Anal. Math. , 1 (1951) pp. 81–115
[10] M.G. Arsove, "The Paley-Wiener theorem in metric linear spaces" Pacific J. Math. , 10 (1960) pp. 365–379
[11] C. Bessaga, A. Pelczyński, "Spaces of continuous functions IV" Studia Math. , 19 (1960) pp. 53–62
[12] R.C. James, "Bases and reflexivity in Banach spaces" Ann. of Math. (2) , 52 : 3 (1950) pp. 518–527
[13] A. Dynin, B. Mityagin, "Criterion for nuclearity in terms of approximate dimension" Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys. , 8 (1960) pp. 535–540
[14] M.M. Day, "Normed linear spaces" , Springer (1958)
[15] A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[16] I.M. Singer, "Bases in Banach spaces" , 1–2 , Springer (1970–1981)
How to Cite This Entry:
Basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basis&oldid=13389
This article was adapted from an original article by M.I. VoitsekhovskiiM.I. Kadets (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article