of a set
A minimal subset that generates it. Generation here means that by application of operations of a certain class to elements it is possible to obtain any element . This concept is related to the concept of dependence: By means of operations from the elements of become dependent on the elements of . Minimality means that no proper subset generates . In a certain sense this property causes the elements of to be independent: None of the elements is generated by the other elements of . For instance, the set of all natural numbers 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 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 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 — (because ).
A basis of the natural numbers of order is a subsequence of the set of natural numbers including 0, which, as a result of -fold addition to itself (the generating operation) yields all of . This means that any natural number can be represented in the form
where . 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 of order 4. In general, the sequence of -th powers of natural numbers is a basis of (Hilbert's theorem), the order of which has been estimated by the Vinogradov method. The concept of a basis of has been generalized to the case of arbitrary sequences of numbers, i.e. functions on .
A set always contains a generating set (in the trivial case: generates ), but minimality may prove to be principally impossible (such a situation is typical of classes 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 is defined as a parametrized set (or population), i.e. as a function on a set of indices with values in , such that ; the cardinality of is sometimes called as the dimension (or rank) of the basis of . For example, a countable everywhere-dense set in a separable topological space may be considered as a basis for it; is generated from 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 (a base) is a basis of the set of all open subsets in ; the generation is effected by taking unions of elements of .
A basis of a Boolean algebra (a dual base of in the sense of Tarski) is a dense set (of minimal cardinality) in ; the generation of from (and hence itself) is determined by the condition (which is equivalent to ), where , , is the unit of and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter as a set such that for an arbitrary there exists an with .
More special cases of bases of a set are introduced according to the following procedure. Let be the Boolean algebra of , i.e. the set of all its subsets. A generating operator (or a closure operator) is a mapping of into itself such that if , then ; ; .
An element is generated by a set if ; in particular, generates if . A minimal set possessing this property is said to be a basis of defined by the operator . A generating operator is of finite type if, for arbitrary and , it follows from that for a certain finite subset ; a generating operator has the property of substitution if, for any and , both and imply that . A generating operator of finite type with the substitution property defines a dependence relation on , i.e. a subdivision of into two classes — dependent and independent sets; a set is said to be dependent if for some , and is said to be independent if for any . Therefore, is dependent (independent) if and only if some (arbitrary) non-empty finite subset(s) is dependent (are independent).
For a set to be a basis of the set it is necessary and sufficient for to be an independent generating set for , or else, a maximal independent set in .
If is an arbitrary independent set, and is an -generating set containing , then there exists a basis in such that . In particular, always has a basis, and any two bases of it have the same cardinality.
In algebraic systems an important role is played by the concept of the so-called free basis , which is characterized by the following property: Any mapping of into any algebraic system (of the same signature) may be extended to a (unique) (homo)morphism from into or, which is the same thing, for any (homo)morphism and any set , the generating operators and satisfy the condition:
An algebraic system with a free basis is said to be free.
A typical example is a basis of a (unitary) module over a ring , that is, a free family of elements from generating . Here, a family of elements of a -module is said to be free if (where for all except a finite number of indices ) implies that for all , and the generation is realized by representing the elements as linear combinations of the elements : There exists a set (dependent on ) of elements such that for all except a finite number of indices , and such that the decomposition
is valid (i.e. is the linear envelope of ). In this sense, the basis is free basis; the converse proposition is also true. Thus, the set of periods of a doubly-periodic function of one complex variable, which is a discrete Abelian group (and hence a module over the ring ), has a free basis, called the period basis of ; 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 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 , considered as a -module.
A basis of a vector space over a field is a (free) basis of the unitary module which underlies . In a similar manner, a basis of an algebra over a field is a basis of the vector space underlying . All bases of a given vector space have the same cardinality, which is equal to the cardinality of ; the latter is called the algebraic dimension of . Each element can be represented as a linear combination of basis elements in a unique way. The elements , which are linear functionals on , are called the components (coordinates) of in the given basis .
A set is a basis in if and only if is a maximal (with respect to inclusion) free set in .
where if is the value of the -th component of in the basis , and 0 otherwise, is called the basis mapping; it is a linear injective mapping of into the space of functions on with values in . In this case the image 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 over a field as a bijective linear mapping from it to some subspace of the space of functions on with values in , where is some suitably chosen set. However, unless additional restrictions (e.g. an order) and additional structures (e.g. a topology) are imposed on , and corresponding compatible conditions on are introduced, the concept of a generalized basis is seldom of use in practice.
A basis of a vector space is sometimes called an algebraic basis; in this way it is stressed that there is no connection with additional structures on , even if they are compatible with its vector structure.
A Hamel basis is a basis of the field of real numbers , considered as a vector space over the field of rational numbers. It was introduced by G. Hamel  to obtain a discontinuous solution of the functional equation ; the graph of its solution is everywhere dense in the plane . To each almost-periodic function corresponds some countable Hamel basis such that each Fourier exponent of this function belongs to the linear envelope of . The elements of may be so chosen that they belong to a sequence ; the set is said to be a basis of the almost-periodic functions. An analogous basis has been constructed in a ring containing a skew-field and which has the unit of 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 over a field ) is a set 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 and makes it possible to obtain, for each element , its decomposition with respect to the basis , which is moreover unique, i.e. a representation of as a limit (in some sense) of linear combinations of elements :
where are linear functionals on with values in , called the components of in the basis , or the coefficients of the decomposition of with respect to the basis . Clearly, for the decomposition of an arbitrary to exist, must be a complete set in , and for such a decomposition to be unique (i.e. for the zero element of to have all components equal to zero), must be a topologically free set in .
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 , called the basis mapping, into some (depending on ) space of functions with values in , defined on a (topological) space , viz.:
where , so that, symbolically, and . Owing to its concrete, effective definition, the structure of is simpler and more illustrative than that of the abstractly given . 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 is substantially smaller, and simplifies at the same time.
The space contains all functions of finite support, and the set of elements of the basis is the bijective inverse image of the set of functions with only one non-zero value which is equal to one:
where if , and if . In other words, is the generator of a one-dimensional subspace which is complementary in to the hyperplane defined by the equation .
Thus, the role of the basis is to organize, out of the set of components which constitute the image of under the basis mapping, a summable (in some sense) set , i.e. a basis "decomposes" a space 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 (-), 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 leads to the concept of the so-called continuous sum of elements from and to corresponding integral representations; the decomposition of the space into (not necessarily one-dimensional) components is used in the spectral theory of linear operators; the consideration of arbitrary topological algebras over a field (e.g. algebras of measures on with values in or even in , algebras of projection operators, etc.) instead of 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 of elements of a space such that each element is in unique correspondence with its series expansion with respect to the basis
which (in the topology of ) converges to . Here, , 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 , , form a basis in the spaces , (periodic functions absolutely summable of degree ); on the contrary, these functions do not form a basis in the spaces , (measurable functions which almost everywhere coincide with bounded functions) or (continuous periodic functions). A necessary, but by far not sufficient, condition for the existence of a countable basis is the separability of (e.g. a countable basis cannot exist in the space of measurable functions on an interval with values in ). Moreover, the space of bounded sequences, not being separable in the topology of , has no countable basis, but the elements , where if , and if , form a basis in the weak topology . The question of the existence of a countable basis in separable Banach spaces (the basis problem) has been negatively solved . The analogous problem for nuclear spaces also has a negative solution .
A countable basis is, however, not always "well-suited" for applications. For example, the components may be discontinuous, the expansion of 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 is not countable, nevertheless the decomposition of with respect to it has a natural definition: the corresponding space consists of functions with countable support. For instance, a complete orthonormal set in a Hilbert space is a basis; if , then (where is the scalar product in ) for all (except possibly a countable set of) indices , and the series converges to . The basis mapping is determined by the orthogonal projections onto the closed subspaces generated by the elements . A basis of the space of all complex-valued almost-periodic functions on consists of the functions ; here, , is the set of countably-valued functions, and the basis mapping is defined by the formula:
An unconditional basis is a countable basis in a space such that the decomposition of any element converges unconditionally (i.e. the sum of the series does not change if an arbitrary number of its terms is rearranged). For instance, in (sequences converging to zero) and (sequences summable of degree , ) the elements form an unconditional basis; in the space of continuous functions on the interval 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 (or, correspondingly, ).
Two bases and of the Banach spaces and , respectively, are said to be equivalent if there exists a bijective linear mapping that can be extended to an isomorphism between and ; 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, 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 of arbitrary cardinality and becoming identical with it if — is a set such that for an arbitrary element there exists a set of linear combinations (partial sums) of elements from , which is called a generalized decomposition of , which is summable to . This means that for any neighbourhood of zero it is possible to find a finite subset such that for any finite set the relation
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 for which the set of semi-norms 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 of zero and for each the family of semi-norms is summable. All unconditional countable bases are absolute, i.e. the series converges for all and all continuous semi-norms . Of all Banach spaces only the space 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 .
A Schauder basis is a basis of a space such that the basis mapping defined by it is continuous (and is therefore an isomorphism onto some space ), i.e. a basis in which the components for any and, in particular, the coefficients of the decomposition of with respect to this basis, are continuous functionals on . This basis was first defined by J. Schauder  for the case . 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 and form a biorthogonal system. Thus, the sequences form countable Schauder bases in the spaces and , . A countable Schauder basis forms a Haar system in the space . In complete metric vector spaces (in particular, in Banach spaces) all countable bases are Schauder bases . In Fréchet spaces the concept of a weak basis and a Schauder basis are identical . In barrelled spaces in which there are no linear continuous functionals, there is also no Schauder basis . However, if a weak Schauder basis exists in these spaces, it is an ordinary Schauder basis . 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 corresponding to it will be a basis in the dual space and will be boundedly complete, i.e. if the boundedness of the set of partial sums of a series implies that this series is convergent . 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 does not contain subspaces isomorphic to (or, respectively, to ).
A Schauder basis in a locally convex space is equicontinuous if for any neighbourhood of zero it is possible to find a neighbourhood of zero such that
for all . 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 . An equicontinuous basis of a nuclear space is absolute.
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Basis. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Basis&oldid=31237