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A [[Locally convex space|locally convex space]] for which all continuous linear mappings into an arbitrary Banach space are nuclear operators (cf. [[Nuclear operator|Nuclear operator]]). The concept of a nuclear space arose [[#References|[1]]] in an investigation of the question: For what spaces are the analogues of Schwartz' kernel theorem valid (see [[Nuclear bilinear form|Nuclear bilinear form]])? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [[#References|[1]]]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear spaces play an important role in the spectral analysis of operators on Hilbert spaces (the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors, etc.) (see [[#References|[2]]]). Nuclear spaces are closely connected with measure theory on locally convex spaces (see [[#References|[3]]]). Nuclear spaces can be characterized in terms of dimension-type invariants (approximative dimension, diametral dimension, etc.) (see [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]). One of these invariants is the functional dimension, which for many spaces consisting of entire analytic functions is the same as the number of variables on which these functions depend (see [[#References|[2]]]).
 
A [[Locally convex space|locally convex space]] for which all continuous linear mappings into an arbitrary Banach space are nuclear operators (cf. [[Nuclear operator|Nuclear operator]]). The concept of a nuclear space arose [[#References|[1]]] in an investigation of the question: For what spaces are the analogues of Schwartz' kernel theorem valid (see [[Nuclear bilinear form|Nuclear bilinear form]])? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [[#References|[1]]]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear spaces play an important role in the spectral analysis of operators on Hilbert spaces (the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors, etc.) (see [[#References|[2]]]). Nuclear spaces are closely connected with measure theory on locally convex spaces (see [[#References|[3]]]). Nuclear spaces can be characterized in terms of dimension-type invariants (approximative dimension, diametral dimension, etc.) (see [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]). One of these invariants is the functional dimension, which for many spaces consisting of entire analytic functions is the same as the number of variables on which these functions depend (see [[#References|[2]]]).
  
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===Examples of nuclear spaces.===
 
===Examples of nuclear spaces.===
  
 +
1) Let  $  {\mathcal E} ( \mathbf R  ^ {n} ) $
 +
be the space of all (real or complex) infinitely-differentiable functions on  $  \mathbf R  ^ {n} $
 +
equipped with the topology of uniform convergence of all derivatives on compact subsets of  $  \mathbf R  ^ {n} $.
 +
The space  $  {\mathcal E}  ^  \prime  ( \mathbf R  ^ {n} ) $
 +
dual to  $  {\mathcal E} ( \mathbf R  ^ {n} ) $
 +
consists of all generalized functions (cf. [[Generalized function|Generalized function]]) with compact support. Let  $  {\mathcal D} ( \mathbf R  ^ {n} ) $
 +
and  $  {\mathcal S} ( \mathbf R  ^ {n} ) $
 +
be the linear subspaces of  $  {\mathcal E} ( \mathbf R  ^ {n} ) $
 +
consisting, respectively, of functions with compact support and of functions that, together with all their derivatives, decrease faster than any power of  $  | x |  ^ {-} 1 $
 +
as  $  | x | \rightarrow \infty $.
 +
The duals  $  {\mathcal D}  ^  \prime  ( \mathbf R  ^ {n} ) $
 +
and  $  {\mathcal S}  ^  \prime  ( \mathbf R  ^ {n} ) $
 +
of  $  {\mathcal D} ( \mathbf R  ^ {n} ) $
 +
and  $  {\mathcal S} ( \mathbf R  ^ {n} ) $,
 +
relative to the standard topology, consist of all generalized functions and of all generalized functions of slow growth, respectively. The spaces  $  {\mathcal E} $,
 +
$  {\mathcal D} $,
 +
$  {\mathcal S} $,
 +
$  {\mathcal E}  ^  \prime  $,
 +
$  {\mathcal D}  ^  \prime  $,
 +
and  $  {\mathcal S}  ^  \prime  $,
 +
equipped with the strong topology, are complete reflexive nuclear spaces.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678601.png" /> be the space of all (real or complex) infinitely-differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678602.png" /> equipped with the topology of uniform convergence of all derivatives on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678603.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678604.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678605.png" /> consists of all generalized functions (cf. [[Generalized function|Generalized function]]) with compact support. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678607.png" /> be the linear subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678608.png" /> consisting, respectively, of functions with compact support and of functions that, together with all their derivatives, decrease faster than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n0678609.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786010.png" />. The duals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786014.png" />, relative to the standard topology, consist of all generalized functions and of all generalized functions of slow growth, respectively. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786020.png" />, equipped with the strong topology, are complete reflexive nuclear spaces.
+
2) let  $  \{ a _ {np} \} $
 
+
be an infinite matrix, where  $  0 \leq  a _ {np} < \infty $
2) let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786021.png" /> be an infinite matrix, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786024.png" />. The space of sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786027.png" />, with the topology defined by the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786028.png" /> (cf. [[Semi-norm|Semi-norm]]), is called a Köthe space, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786029.png" />. This space is nuclear if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786030.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786032.png" />.
+
and  $  a _ {np} \leq  a _ {n ( p + 1) }  $,  
 +
n, p = 1, 2 , . . . $.  
 +
The space of sequences $  \xi = \{ \xi _ {n} \} $
 +
for which $  | \xi | _ {p} = \sum _ {n = 0 }  ^  \infty  | \xi _ {n} | a _ {np} < \infty $
 +
for all $  p $,  
 +
with the topology defined by the semi-norms $  \xi \rightarrow | \xi | _ {p} $(
 +
cf. [[Semi-norm|Semi-norm]]), is called a Köthe space, and is denoted by $  {\mathcal K} ( a _ {np} ) $.  
 +
This space is nuclear if and only if for any $  p $
 +
one can find a $  q $
 +
such that $  \sum _ {n = 0 }  ^  \infty  ( a _ {np} /a _ {nq} ) < \infty $.
  
 
==Heredity properties.==
 
==Heredity properties.==
 
A locally convex space is nuclear if and only if its completion is nuclear. Every subspace (separable quotient space) of a nuclear space is nuclear. The direct sum, the inductive limit of a countable family of nuclear spaces, and also the product and the projective limit of any family of nuclear spaces, is again nuclear.
 
A locally convex space is nuclear if and only if its completion is nuclear. Every subspace (separable quotient space) of a nuclear space is nuclear. The direct sum, the inductive limit of a countable family of nuclear spaces, and also the product and the projective limit of any family of nuclear spaces, is again nuclear.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786033.png" /> be an arbitrary locally convex space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786034.png" /> denote its dual equipped with the strong topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786035.png" /> is nuclear, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786036.png" /> is called conuclear. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786037.png" /> is arbitrary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786038.png" /> is a nuclear space, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786039.png" /> of continuous linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786040.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786041.png" /> is nuclear with respect to the strong [[Operator topology|operator topology]] (simple convergence); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786042.png" /> is semi-reflexive and conuclear, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786043.png" /> is nuclear also in the topology of bounded convergence.
+
Let $  E $
 +
be an arbitrary locally convex space, and let $  E  ^  \prime  $
 +
denote its dual equipped with the strong topology. If $  E  ^  \prime  $
 +
is nuclear, then $  E $
 +
is called conuclear. If $  E $
 +
is arbitrary and $  F $
 +
is a nuclear space, then the space $  L ( E, F  ) $
 +
of continuous linear operators from $  E $
 +
into $  F $
 +
is nuclear with respect to the strong [[Operator topology|operator topology]] (simple convergence); if $  E $
 +
is semi-reflexive and conuclear, then $  L ( E, F  ) $
 +
is nuclear also in the topology of bounded convergence.
  
 
==Metric and dually-metric nuclear spaces.==
 
==Metric and dually-metric nuclear spaces.==
A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786044.png" /> is called dually metric, or a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786046.png" />, if it has a countable fundamental system of bounded sets and if every (strongly) bounded countable union of equicontinuous subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786047.png" /> is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]). Any strong dual of a metrizable locally convex space is dually metric; the converse is not true. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786048.png" /> is a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786050.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786051.png" /> (a Fréchet space, that is, complete and metrizable). Examples of nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786053.png" /> are Köthe spaces, and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786055.png" />; accordingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786057.png" /> are nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786058.png" />. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786060.png" /> are neither metric nor dually metric.
+
A locally convex space $  E $
 +
is called dually metric, or a space of type $  ( {\mathcal D} {\mathcal F} ) $,  
 +
if it has a countable fundamental system of bounded sets and if every (strongly) bounded countable union of equicontinuous subsets in $  E  ^  \prime  $
 +
is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]). Any strong dual of a metrizable locally convex space is dually metric; the converse is not true. If $  E $
 +
is a space of type $  ( {\mathcal D} {\mathcal F} ) $,  
 +
then $  E  ^  \prime  $
 +
is of type $  ( {\mathcal F} ) $(
 +
a Fréchet space, that is, complete and metrizable). Examples of nuclear spaces of type $  ( {\mathcal F} ) $
 +
are Köthe spaces, and also $  {\mathcal E} $
 +
and $  {\mathcal S} $;  
 +
accordingly, $  {\mathcal E}  ^  \prime  $
 +
and $  {\mathcal S}  ^  \prime  $
 +
are nuclear spaces of type $  ( {\mathcal D} {\mathcal F} ) $.  
 +
The spaces $  {\mathcal D} $
 +
and $  {\mathcal D}  ^  \prime  $
 +
are neither metric nor dually metric.
  
Metric and dually-metric nuclear spaces are separable, and if complete, they are reflexive. The transition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786061.png" /> to the dual space establishes a one-to-one correspondence between nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786062.png" /> and complete nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786063.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786064.png" /> is a complete nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786065.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786066.png" /> is a nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786067.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786068.png" />, equipped with the topology of bounded convergence, is nuclear and conuclear.
+
Metric and dually-metric nuclear spaces are separable, and if complete, they are reflexive. The transition $  E \rightarrow E  ^  \prime  $
 +
to the dual space establishes a one-to-one correspondence between nuclear spaces of type $  ( {\mathcal F} ) $
 +
and complete nuclear spaces of type $  ( {\mathcal D} {\mathcal F} ) $.  
 +
If $  E $
 +
is a complete nuclear space of type $  ( {\mathcal D} {\mathcal F} ) $
 +
and if $  F $
 +
is a nuclear space of type $  ( {\mathcal F} ) $,  
 +
then $  L ( E, F  ) $,  
 +
equipped with the topology of bounded convergence, is nuclear and conuclear.
  
Every nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786069.png" /> is isomorphic to a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786070.png" /> of infinitely-differentiable functions on the real line, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786071.png" /> is a universal space for the nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786072.png" /> (see [[#References|[10]]]). A Fréchet space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786073.png" /> is nuclear if and only if every unconditionally-convergent series (cf. [[Unconditional convergence|Unconditional convergence]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786074.png" /> is absolutely convergent (that is, with respect to any continuous semi-norm). Spaces of holomorphic functions on nuclear spaces of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786076.png" /> have been studied intensively (see [[#References|[11]]]).
+
Every nuclear space of type $  ( {\mathcal F} ) $
 +
is isomorphic to a subspace of the space $  {\mathcal E} ( \mathbf R ) $
 +
of infinitely-differentiable functions on the real line, that is, $  {\mathcal E} ( \mathbf R ) $
 +
is a universal space for the nuclear spaces of type $  ( {\mathcal F} ) $(
 +
see [[#References|[10]]]). A Fréchet space $  E $
 +
is nuclear if and only if every unconditionally-convergent series (cf. [[Unconditional convergence|Unconditional convergence]]) in $  E $
 +
is absolutely convergent (that is, with respect to any continuous semi-norm). Spaces of holomorphic functions on nuclear spaces of types $  ( {\mathcal F} ) $
 +
and $  ( {\mathcal D} {\mathcal F} ) $
 +
have been studied intensively (see [[#References|[11]]]).
  
 
==Tensor products of nuclear spaces, and spaces of vector functions.==
 
==Tensor products of nuclear spaces, and spaces of vector functions.==
The algebraic tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786077.png" /> of two locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786079.png" /> can be equipped with the projective and injective topologies, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786080.png" /> becomes a topological tensor product. The projective topology is the strongest locally convex topology in which the canonical bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786081.png" /> is continuous. The injective topology (or the topology of (bi) equicontinuous convergence) is induced by the natural imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786083.png" /> is the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786084.png" /> equipped with the [[Mackey topology|Mackey topology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786085.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786086.png" /> is the space of continuous linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786087.png" /> equipped with the topology of [[Uniform convergence|uniform convergence]] on equicontinuous sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786088.png" />. Under this imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786089.png" /> goes into the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786091.png" /> denotes the value of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786092.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786093.png" />. The completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786094.png" /> in the projective (respectively, injective) topology is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786095.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786096.png" />).
+
The algebraic tensor product $  E \otimes F $
 +
of two locally convex spaces $  E $
 +
and $  F $
 +
can be equipped with the projective and injective topologies, and then $  E \otimes F $
 +
becomes a topological tensor product. The projective topology is the strongest locally convex topology in which the canonical bilinear mapping $  E \times F \rightarrow E \otimes F $
 +
is continuous. The injective topology (or the topology of (bi) equicontinuous convergence) is induced by the natural imbedding $  E \otimes F \rightarrow L _ {e} ( E _  \tau  ^  \prime  , F  ) $,  
 +
where $  E _  \tau  ^  \prime  $
 +
is the dual of $  E $
 +
equipped with the [[Mackey topology|Mackey topology]] $  \tau ( E  ^  \prime  , E) $,  
 +
and $  L _ {e} ( E _  \tau  ^  \prime  , F  ) $
 +
is the space of continuous linear mappings $  E _  \tau  ^  \prime  \rightarrow F $
 +
equipped with the topology of [[Uniform convergence|uniform convergence]] on equicontinuous sets in $  E  ^  \prime  $.  
 +
Under this imbedding $  x \otimes y \in E \otimes F $
 +
goes into the operator $  x  ^  \prime  \mapsto \langle  x, x  ^  \prime  \rangle y $,  
 +
where $  \langle  x, x  ^  \prime  \rangle $
 +
denotes the value of the functional $  x  ^  \prime  \in E  ^  \prime  $
 +
at $  x \in E $.  
 +
The completion of $  E \otimes F $
 +
in the projective (respectively, injective) topology is denoted by $  E \widehat \otimes  F $(
 +
respectively, $  E \widetilde \otimes  F  $).
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786097.png" /> to be a nuclear space it is necessary and sufficient that for any locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786098.png" /> the projective and injective topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n06786099.png" /> coincide, that is,
+
For $  E $
 +
to be a nuclear space it is necessary and sufficient that for any locally convex space $  F $
 +
the projective and injective topologies in $  E \otimes F $
 +
coincide, that is,
  
<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/n/n067/n067860/n067860100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
E \widehat \otimes  F  = E \widetilde \otimes  F.
 +
$$
  
Actually, it suffices to require that (1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860101.png" />, the space of summable sequences, or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860102.png" /> equal to a fixed space with an unconditional basis (see [[#References|[12]]]). Nevertheless, there is a (non-nuclear) infinite-dimensional separable Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860104.png" /> (see [[#References|[13]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860106.png" /> are complete spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860107.png" /> is nuclear, then the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860108.png" /> can be extended to an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860110.png" />.
+
Actually, it suffices to require that (1) holds for $  F = l _ {1} $,  
 +
the space of summable sequences, or for $  F $
 +
equal to a fixed space with an unconditional basis (see [[#References|[12]]]). Nevertheless, there is a (non-nuclear) infinite-dimensional separable Banach space $  X $
 +
such that $  X \widehat \otimes  X = X \widetilde \otimes  X $(
 +
see [[#References|[13]]]). If $  E $
 +
and $  F $
 +
are complete spaces and $  F $
 +
is nuclear, then the imbedding $  E \otimes F \rightarrow L _ {e} ( E _  \tau  ^  \prime  , F  ) $
 +
can be extended to an isomorphism between $  E \widehat \otimes  F $
 +
and $  L _ {e} ( E _  \tau  ^  \prime  , F  ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860111.png" /> is a non-null nuclear space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860112.png" /> is nuclear if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860113.png" /> is nuclear. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860115.png" /> are both spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860116.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860117.png" />) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860118.png" /> is nuclear, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860119.png" />.
+
If $  E $
 +
is a non-null nuclear space, then $  E \widehat \otimes  F $
 +
is nuclear if and only if $  F $
 +
is nuclear. If $  E $
 +
and $  F $
 +
are both spaces of type $  ( {\mathcal F} ) $(
 +
or $  ( {\mathcal D} {\mathcal F} ) $)  
 +
and if $  E $
 +
is nuclear, then $  ( E \widehat \otimes  F  )  ^  \prime  = E  ^  \prime  \widehat \otimes  F  ^  \prime  $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860120.png" /> be a complete nuclear space consisting of scalar functions (not all) on a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860121.png" />; let also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860122.png" /> be the inductive limit (locally convex hull) of a countable sequence of spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860123.png" />, and let the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860124.png" /> be not weaker than the topology of pointwise convergence of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860125.png" />. Then for any complete space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860126.png" /> one can identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860127.png" /> with the space of all mappings (vector functions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860128.png" /> for which the scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860129.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860130.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860131.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860132.png" /> coincides with the space of all infinitely-differentiable vector functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860133.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860134.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860135.png" />.
+
Let $  E $
 +
be a complete nuclear space consisting of scalar functions (not all) on a certain set $  T $;  
 +
let also $  E $
 +
be the inductive limit (locally convex hull) of a countable sequence of spaces of type $  ( {\mathcal F} ) $,  
 +
and let the topology on $  E $
 +
be not weaker than the topology of pointwise convergence of functions on $  T $.  
 +
Then for any complete space $  F $
 +
one can identify $  E \widehat \otimes  F $
 +
with the space of all mappings (vector functions) $  T \rightarrow F $
 +
for which the scalar function $  t \mapsto \langle  f ( t), y  ^  \prime  \rangle $
 +
belongs to $  E $
 +
for all $  y  ^  \prime  \in F ^ { \prime } $.  
 +
In particular, $  {\mathcal E} ( \mathbf R  ^ {n} ) \widehat \otimes  F $
 +
coincides with the space of all infinitely-differentiable vector functions on $  \mathbf R  ^ {n} $
 +
with values in $  F $,  
 +
and $  {\mathcal E} ( \mathbf R  ^ {n} ) \widehat \otimes  {\mathcal E} ( \mathbf R  ^ {m} ) = {\mathcal E} ( \mathbf R  ^ {n} \times \mathbf R  ^ {m} ) = {\mathcal E} ( \mathbf R ^ {n + m } ) $.
  
 
==The structure of nuclear spaces.==
 
==The structure of nuclear spaces.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860136.png" /> be a convex circled (i.e. convex balanced) neighbourhood of zero in a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860137.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860138.png" /> be the Minkowski functional (continuous semi-norm) corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860139.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860140.png" /> be the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860141.png" /> with the norm induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860142.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860143.png" /> be the completion of the normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860144.png" />. There is defined a continuous canonical linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860145.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860146.png" /> contains a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860147.png" />, then the continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860148.png" /> is defined canonically.
+
Let $  U $
 +
be a convex circled (i.e. convex balanced) neighbourhood of zero in a locally convex space $  E $,  
 +
and let $  p $
 +
be the Minkowski functional (continuous semi-norm) corresponding to $  U $.  
 +
Let $  E _ {U} $
 +
be the quotient space $  E/p  ^ {-} 1 ( 0) $
 +
with the norm induced by $  p $,  
 +
and let $  {E _ {U} } hat $
 +
be the completion of the normed space $  E _ {U} $.  
 +
There is defined a continuous canonical linear mapping $  E \rightarrow {E _ {U} } hat $;  
 +
if $  U $
 +
contains a neighbourhood $  V $,  
 +
then the continuous linear mapping $  {E _ {V} } hat \rightarrow {E _ {U} } hat $
 +
is defined canonically.
  
For a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860149.png" /> the following conditions are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860150.png" /> is nuclear; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860151.png" /> has a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860152.png" /> of convex circled neighbourhoods of zero such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860153.png" /> the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860154.png" /> is a nuclear operator; 3) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860155.png" /> is nuclear for any convex circled neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860156.png" /> of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860157.png" />; and 4) every convex circled neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860158.png" /> of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860159.png" /> contains another such neighbourhood of zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860160.png" />, such that the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860161.png" /> is nuclear.
+
For a locally convex space $  E $
 +
the following conditions are equivalent: 1) $  E $
 +
is nuclear; 2) $  E $
 +
has a basis $  \mathfrak B $
 +
of convex circled neighbourhoods of zero such that for any $  U \in \mathfrak B $
 +
the canonical mapping $  E \rightarrow {E _ {U} } hat $
 +
is a nuclear operator; 3) the mapping $  E \rightarrow {E _ {U} } hat $
 +
is nuclear for any convex circled neighbourhood $  U $
 +
of zero in $  E $;  
 +
and 4) every convex circled neighbourhood $  U $
 +
of zero in $  E $
 +
contains another such neighbourhood of zero, $  V $,  
 +
such that the canonical mapping $  {E _ {V} } hat \rightarrow {E _ {U} } hat $
 +
is nuclear.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860162.png" /> be a nuclear space. For any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860163.png" /> of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860164.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860165.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860166.png" /> there is a convex circled neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860167.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860168.png" /> is (norm) isomorphic to a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860169.png" /> of sequences with summable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860170.png" />-th powers. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860171.png" /> coincides with a subspace of the projective limit of a family of spaces isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860172.png" />. In particular (the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860173.png" />), in any nuclear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860174.png" /> there is a basis of neighbourhoods of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860175.png" /> such that all the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860176.png" /> are Hilbert spaces; thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860177.png" /> is Hilbertian, that is, the topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860178.png" /> can be generated by a family of semi-norms each of which is obtained from a certain non-negative definite Hermitian form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860179.png" />. Any complete nuclear space is isomorphic to the projective limit of a family of Hilbert spaces. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860180.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860181.png" /> is nuclear if and only if it can be represented as the projective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860182.png" /> of a countable family of Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860183.png" />, such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860184.png" /> are nuclear operators (or, at least, Hilbert–Schmidt operators, cf. [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860185.png" />.
+
Let $  E $
 +
be a nuclear space. For any neighbourhood $  U $
 +
of zero in $  E $
 +
and for any $  q $
 +
such that $  1 \leq  q \leq  \infty $
 +
there is a convex circled neighbourhood $  V \subset  U $
 +
for which $  E _ {V} $
 +
is (norm) isomorphic to a subspace of the space $  l _ {q} $
 +
of sequences with summable $  q $-
 +
th powers. Thus, $  E $
 +
coincides with a subspace of the projective limit of a family of spaces isomorphic to $  l _ {q} $.  
 +
In particular (the case $  q = 2 $),  
 +
in any nuclear space $  E $
 +
there is a basis of neighbourhoods of zero $  \{ U _  \alpha  \} $
 +
such that all the spaces $  {E _ {U _  \alpha  } } hat $
 +
are Hilbert spaces; thus, $  E $
 +
is Hilbertian, that is, the topology in $  E $
 +
can be generated by a family of semi-norms each of which is obtained from a certain non-negative definite Hermitian form on $  E \times E $.  
 +
Any complete nuclear space is isomorphic to the projective limit of a family of Hilbert spaces. A space $  E $
 +
of type $  ( {\mathcal F} ) $
 +
is nuclear if and only if it can be represented as the projective limit $  E = \lim\limits _  \leftarrow  g _ {mn} H _ {n} $
 +
of a countable family of Hilbert spaces $  H _ {n} $,  
 +
such that the $  g _ {mn} $
 +
are nuclear operators (or, at least, Hilbert–Schmidt operators, cf. [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) for $  m < n $.
  
 
==Bases in nuclear spaces.==
 
==Bases in nuclear spaces.==
In a nuclear space every equicontinuous basis is absolute. In a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860186.png" /> any countable basis (even if weak) is an equicontinuous Schauder basis (cf. [[Basis|Basis]]), so that in a nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860187.png" /> any basis is absolute (in particular, unconditional). A similar result holds for complete nuclear spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860188.png" />, and for all nuclear spaces for which the [[Closed-graph theorem|closed-graph theorem]] holds. A quotient space of a nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860189.png" /> with a basis does not necessarily have a basis (see [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]).
+
In a nuclear space every equicontinuous basis is absolute. In a space of type $  ( {\mathcal F} ) $
 +
any countable basis (even if weak) is an equicontinuous Schauder basis (cf. [[Basis|Basis]]), so that in a nuclear space of type $  ( {\mathcal F} ) $
 +
any basis is absolute (in particular, unconditional). A similar result holds for complete nuclear spaces of type $  ( {\mathcal D} {\mathcal F} ) $,  
 +
and for all nuclear spaces for which the [[Closed-graph theorem|closed-graph theorem]] holds. A quotient space of a nuclear space of type $  ( {\mathcal F} ) $
 +
with a basis does not necessarily have a basis (see [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860190.png" /> be a nuclear space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860191.png" />. A topology can be defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860192.png" /> by a countable system of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860194.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860195.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860196.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860197.png" /> has a basis or a continuous norm, then the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860198.png" /> can be taken as norms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860199.png" /> be a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860200.png" />; then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860201.png" /> can be expressed as an (absolutely and unconditionally) convergent series
+
Let $  E $
 +
be a nuclear space of type $  ( {\mathcal F} ) $.  
 +
A topology can be defined in $  E $
 +
by a countable system of semi-norms $  x \mapsto \| x \| _ {q} $,
 +
$  q = 1, 2 \dots $
 +
where $  \| x \| _ {q} \leq  \| x \| _ {q + 1 }  $
 +
for all $  x \in E $.  
 +
If $  E $
 +
has a basis or a continuous norm, then the semi-norms $  \| \cdot \| $
 +
can be taken as norms. Let $  \{ e _ {n} \} $
 +
be a basis in $  E $;  
 +
then any $  x \in E $
 +
can be expressed as an (absolutely and unconditionally) convergent series
  
<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/n/n067/n067860/n067860202.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {n = 1 } ^  \infty 
 +
\xi _ {n} e _ {n} ,
 +
$$
  
where the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860203.png" /> have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860204.png" />, and the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860205.png" /> form a bi-orthogonal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860206.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860207.png" /> is isomorphic to the Köthe space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860208.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860209.png" />; under this isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860210.png" /> goes into the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860211.png" /> of its coordinates. A basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860212.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860213.png" /> is equivalent to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860214.png" /> (that is, it can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860215.png" /> by an isomorphism) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860216.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860217.png" /> coincide as sets [[#References|[4]]]. A basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860218.png" /> is called regular (or proper) if there is a system of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860219.png" /> and a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860220.png" /> of indices such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860221.png" /> is monotone decreasing for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860222.png" />. If a nuclear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860223.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860224.png" /> has a regular basis, then any two bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860225.png" /> are quasi-equivalent (that is, they can be made equivalent by a permutation and a normalization of the elements of one of them). There are other sufficient conditions for all bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860226.png" /> to be quasi-equivalent (see [[#References|[4]]], [[#References|[14]]]). A complete description of the class of nuclear spaces with this property is not known (1984).
+
where the coordinates $  \xi _ {n} $
 +
have the form $  \xi _ {n} = \langle  x, x _ {n}  ^  \prime  \rangle $,  
 +
and the functionals $  x _ {n}  ^  \prime  $
 +
form a bi-orthogonal basis in $  E  ^  \prime  $.  
 +
$  E $
 +
is isomorphic to the Köthe space $  {\mathcal K} ( a _ {nq} ) $,  
 +
where $  a _ {nq} = \| e _ {n} \| _ {q} $;  
 +
under this isomorphism $  x \in E $
 +
goes into the sequence $  \{ \xi _ {n} \} $
 +
of its coordinates. A basis $  \{ f _ {n} \} $
 +
in $  E $
 +
is equivalent to the basis $  \{ e _ {n} \} $(
 +
that is, it can be obtained from $  \{ e _ {n} \} $
 +
by an isomorphism) if and only if $  {\mathcal K} ( \| e _ {n} \| _ {q} ) $
 +
and $  {\mathcal K} ( \| f _ {n} \| _ {q} ) $
 +
coincide as sets [[#References|[4]]]. A basis $  \{ f _ {n} \} $
 +
is called regular (or proper) if there is a system of norms $  \| \cdot \| _ {q} $
 +
and a permutation $  \sigma $
 +
of indices such that $  \| f _ {\sigma ( n) }  \| _ {q} / \|  f _ {\sigma ( n) }  \| _ {r} $
 +
is monotone decreasing for all $  r \geq  q $.  
 +
If a nuclear space $  E $
 +
of type $  ( {\mathcal F} ) $
 +
has a regular basis, then any two bases in $  E $
 +
are quasi-equivalent (that is, they can be made equivalent by a permutation and a normalization of the elements of one of them). There are other sufficient conditions for all bases in $  E $
 +
to be quasi-equivalent (see [[#References|[4]]], [[#References|[14]]]). A complete description of the class of nuclear spaces with this property is not known (1984).
  
Example. The Hermite functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860227.png" /> form a basis in the complete metric nuclear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860228.png" /> of smooth functions on the real line that are rapidly decreasing together with all their derivatives. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860229.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860230.png" />.
+
Example. The Hermite functions $  \phi _ {n} ( t) = e ^ {t  ^ {2} /2 } ( {d  ^ {n} } / {dt  ^ {n} } ) ( e ^ {- t  ^ {2} } ) $
 +
form a basis in the complete metric nuclear space $  {\mathcal S} ( \mathbf R ) $
 +
of smooth functions on the real line that are rapidly decreasing together with all their derivatives. $  {\mathcal S} ( \mathbf R ) $
 +
is isomorphic to $  {\mathcal K} ( n ^ {p} ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.A. Minlos,  "Generalized random processes and their extension in measure"  ''Trudy Moskov. Mat. Obshch.'' , '''8'''  (1959)  pp. 497–518  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.S. Mityagin,  "Approximate dimension and bases in nuclear spaces"  ''Russian Math. Surveys'' , '''16''' :  4  pp. 59–127  ''Uspekhi Mat. Nauk'' , '''16''' :  4  (1961)  pp. 63–132</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Dubinsky,  "Structure of nuclear Fréchet spaces" , Springer  (1979)</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"  ''Funct. Anal. Appl.'' , '''8''' :  4  (1974)  pp. 304–313  ''Funktsional. Anal. i Prilozhen.'' , '''8''' :  4  (1974)  pp. 35–47</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Atzmon,  "An operator without invariant subspaces on a nuclear Fréchet space"  ''Ann. of Math.'' , '''117''' :  3  (1983)  pp. 669–694</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  T. Komura,  Y. Komura,  "Ueber die Einbettung der nuklearen Räume in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860231.png" />"  ''Math. Ann.'' , '''162'''  (1965–1966)  pp. 284–288</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  S. Dineen,  "Complex analysis in locally convex spaces" , North-Holland  (1981)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  K. John,  V. Zizler,  "On a tensor product characterization of nuclearity"  ''Math. Ann.'' , '''244''' :  1  (1979)  pp. 83–87</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  G. Pisier,  "Contre-example à une conjecture de Grothendieck"  ''C.R. Acad. Sci. Paris'' , '''293'''  (1981)  pp. 681–683  (English abstract)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M.M. Dragilev,  "Bases in Köthe spaces" , Rostov-on-Don  (1983)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.A. Minlos,  "Generalized random processes and their extension in measure"  ''Trudy Moskov. Mat. Obshch.'' , '''8'''  (1959)  pp. 497–518  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.S. Mityagin,  "Approximate dimension and bases in nuclear spaces"  ''Russian Math. Surveys'' , '''16''' :  4  pp. 59–127  ''Uspekhi Mat. Nauk'' , '''16''' :  4  (1961)  pp. 63–132</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Dubinsky,  "Structure of nuclear Fréchet spaces" , Springer  (1979)</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"  ''Funct. Anal. Appl.'' , '''8''' :  4  (1974)  pp. 304–313  ''Funktsional. Anal. i Prilozhen.'' , '''8''' :  4  (1974)  pp. 35–47</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Atzmon,  "An operator without invariant subspaces on a nuclear Fréchet space"  ''Ann. of Math.'' , '''117''' :  3  (1983)  pp. 669–694</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  T. Komura,  Y. Komura,  "Ueber die Einbettung der nuklearen Räume in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860231.png" />"  ''Math. Ann.'' , '''162'''  (1965–1966)  pp. 284–288</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  S. Dineen,  "Complex analysis in locally convex spaces" , North-Holland  (1981)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  K. John,  V. Zizler,  "On a tensor product characterization of nuclearity"  ''Math. Ann.'' , '''244''' :  1  (1979)  pp. 83–87</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  G. Pisier,  "Contre-example à une conjecture de Grothendieck"  ''C.R. Acad. Sci. Paris'' , '''293'''  (1981)  pp. 681–683  (English abstract)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M.M. Dragilev,  "Bases in Köthe spaces" , Rostov-on-Don  (1983)  (In Russian)</TD></TR></table>
  
 +
====Comments====
 +
A generalized function is also called a distribution, and a generalized function of slow growth is also called a tempered distribution.
  
 +
Let  $  F $
 +
be a topological linear space,  $  U $
 +
a neighbourhood of zero in  $  F $,
 +
$  A $
 +
a set in  $  F $,
 +
and  $  \epsilon $
 +
a (small) positive number. An  $  \epsilon $-
 +
set for  $  A $
 +
relative to a neighbourhood  $  U $
 +
of zero is a set  $  B $
 +
such that for every  $  a \in A $
 +
there is a  $  b \in B $
 +
such that  $  a \in b + \epsilon U $.
 +
Let  $  N ( \epsilon , A , U ) $
 +
be the smallest number of elements in  $  \epsilon $-
 +
sets for  $  A $
 +
relative to  $  U $.
 +
The functional dimension of  $  F $
 +
is defined by
  
====Comments====
+
$$
A generalized function is also called a distribution, and a generalized function of slow growth is also called a tempered distribution.
+
\mathop{\rm df} ( F  )  = \sup _ { U }  \inf _ { V } \
 +
{\lim\limits  \sup } _ {\epsilon \rightarrow 0 } \
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860232.png" /> be a topological linear space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860233.png" /> a neighbourhood of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860235.png" /> a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860236.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860237.png" /> a (small) positive number. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860239.png" />-set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860240.png" /> relative to a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860241.png" /> of zero is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860242.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860243.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860244.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860245.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860246.png" /> be the smallest number of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860247.png" />-sets for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860248.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860249.png" />. The functional dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860250.png" /> is defined by
+
\frac{ { \mathop{\rm ln}  \mathop{\rm ln} }  N ( \epsilon , V , U ) }{ { \mathop{\rm ln}  \mathop{\rm ln} }  \epsilon  ^ {-} 1 }
 +
,
 +
$$
  
<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/n/n067/n067860/n067860251.png" /></td> </tr></table>
+
where  $  U , V $
 +
range over the neighbourhoods of zero in  $  F $.  
 +
Cf. [[#References|[2]]], Sect. I.3.8 for more details.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860252.png" /> range over the neighbourhoods of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860253.png" />. Cf. [[#References|[2]]], Sect. I.3.8 for more details.
+
Let  $  F $
 +
be a locally convex space and consider two neighbourhoods of zero $  U , V $
 +
such that  $  U $
 +
absorbs  $  V $,  
 +
i.e. $  V \subset  \rho U $
 +
for some positive number  $  \rho $.  
 +
Let
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860254.png" /> be a locally convex space and consider two neighbourhoods of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860255.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860256.png" /> absorbs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860257.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860258.png" /> for some positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860259.png" />. Let
+
$$
 +
\delta _ {r} ( U , V )  = \inf \{ \delta : \exists \
 +
\textrm{ subspace }  G  \textrm{ of dimension  }  \leq  r
 +
$$
  
<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/n/n067/n067860/n067860260.png" /></td> </tr></table>
+
$$
 +
{}  \textrm{ such  that  }  V \subset  \delta U + G \} .
 +
$$
  
<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/n/n067/n067860/n067860261.png" /></td> </tr></table>
+
This number is called the  $  r $-
 +
th diameter of  $  V $
 +
with respect to  $  U $.
 +
The diametral dimension of a locally convex space is the collection of all sequences  $  ( d _ {r} ) _ {r \in \mathbf N \cup \{ 0 \} }  $
 +
of non-negative numbers with the property that for each neighbourhood of zero  $  U $
 +
there is a neighbourhood  $  V $
 +
of zero absorbed by  $  U $
 +
for which  $  \delta _ {r} ( U , V ) \leq  d _ {r} $,
 +
$  r \in \mathbf N \cup \{ 0 \} $.
  
This number is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860263.png" />-th diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860264.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860265.png" />. The diametral dimension of a locally convex space is the collection of all sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860266.png" /> of non-negative numbers with the property that for each neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860267.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860268.png" /> of zero absorbed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860269.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860270.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860271.png" />.
+
A locally convex space  $  E $
 +
is nuclear if and only if for some (respectively, each) positive number $  \lambda $
 +
the sequence  $  ( ( r + 1 ) ^ {- \lambda } ) _ {r \in N \cup \{ 0 \} }  $
 +
belongs to the diametral dimension of $  E $.  
 +
See [[#References|[5]]], Chapt. 9 for more details.
  
A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860272.png" /> is nuclear if and only if for some (respectively, each) positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860273.png" /> the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860274.png" /> belongs to the diametral dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860275.png" />. See [[#References|[5]]], Chapt. 9 for more details.
+
Let again  $  U , V $
 +
be neighbourhoods of zero of a locally convex space $  F $
 +
such that  $  U $
 +
absorbs  $  V $.  
 +
The  $  \epsilon $-
 +
content of  $  V $
 +
with respect to  $  U $
 +
is the supremum  $  M _  \epsilon  ( U , V ) $
 +
of all natural numbers  $  m $
 +
such that there are  $  x _ {1} \dots x _ {m} \in V $
 +
with  $  x _ {1} \dots x _ {k} \notin \epsilon U $
 +
for all  $  i \neq k $.  
 +
The approximative dimension of a locally convex space  $  F $
 +
is the collection of all positive functions  $  \phi $
 +
on  $  ( 0 , \infty ) $
 +
such that for each neighbourhood  $  U $
 +
of zero there is a neighbourhood  $  V $
 +
of zero absorbed by  $  U $
 +
such that
  
Let again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860276.png" /> be neighbourhoods of zero of a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860277.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860278.png" /> absorbs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860279.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860281.png" />-content of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860282.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860283.png" /> is the supremum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860284.png" /> of all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860285.png" /> such that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860286.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860287.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860288.png" />. The approximative dimension of a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860289.png" /> is the collection of all positive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860290.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860291.png" /> such that for each neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860292.png" /> of zero there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860293.png" /> of zero absorbed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860294.png" /> such that
+
$$
 +
\lim\limits _ {\epsilon \rightarrow 0 }  \phi ( \epsilon )  ^ {-} 1
 +
M _  \epsilon  ( U , V )  = 0 .
 +
$$
  
<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/n/n067/n067860/n067860295.png" /></td> </tr></table>
+
The number  $  \rho ( U , V ) $
 +
is defined by the exponential rate of growth of  $  M _  \epsilon  ( U , V ) $
 +
as  $  \epsilon \rightarrow 0 $.  
 +
More precisely,
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860296.png" /> is defined by the exponential rate of growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860297.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860298.png" />. More precisely,
+
$$
 +
\rho ( U , V )  = {\lim\limits  \sup } _ {\epsilon \rightarrow 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/n/n067/n067860/n067860299.png" /></td> </tr></table>
+
\frac{ { \mathop{\rm ln}  \mathop{\rm ln} }  M _  \epsilon  ( U , V ) }{ \mathop{\rm ln}  \epsilon  ^ {-} 1 }
 +
.
 +
$$
  
A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860300.png" /> is nuclear if and only if for some (respectively, each) positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860301.png" /> the following condition is satisfied: For each neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860302.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860303.png" /> of zero absorbed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860304.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860305.png" />. Cf. [[#References|[5]]], Chapt. 9 for more details.
+
A locally convex space $  E $
 +
is nuclear if and only if for some (respectively, each) positive number $  \rho $
 +
the following condition is satisfied: For each neighbourhood of zero $  U $
 +
there is a neighbourhood $  V $
 +
of zero absorbed by $  U $
 +
such that $  \rho ( U , V ) \leq  \rho $.  
 +
Cf. [[#References|[5]]], Chapt. 9 for more details.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860306.png" /> be a bounded circled neighbourhood of a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860307.png" />. The Minkowski functional associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860308.png" /> is defined by
+
Let $  U $
 +
be a bounded circled neighbourhood of a topological vector space $  F $.  
 +
The Minkowski functional associated to $  U $
 +
is defined by
  
<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/n/n067/n067860/n067860309.png" /></td> </tr></table>
+
$$
 +
q ( x)  = \inf _ {x \in \alpha U }  \alpha ,\ \
 +
\alpha \geq  0 .
 +
$$
  
This is well-defined for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860310.png" /> since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860311.png" /> is absorbent (i.e. for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860312.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860313.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067860/n067860314.png" />). Cf. [[#References|[a7]]], Sects. 15.10, 16.4.
+
This is well-defined for each $  x $
 +
since $  U $
 +
is absorbent (i.e. for each $  x \in F $
 +
there is an $  \alpha $
 +
such that $  x \in \alpha U $).  
 +
Cf. [[#References|[a7]]], Sects. 15.10, 16.4.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Grothendieck,  "Résumé de la théorie métrique des produits tensoriels topologiques"  ''Bol. Soc. Mat. Sao-Paulo'' , '''8'''  (1956)  pp. 1–79</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,  "Topological vector spaces" , Gordon &amp; Breach  (1973)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Pisier,  "Factorization of linear operators and geometry of Banach spaces" , Amer. Math. Soc.  (1986)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Pisier,  "Counterexamples to a conjecture of Grothendieck"  ''Acta. Math.'' , '''151'''  (1983)  pp. 181–208</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.F. Colombeau,  "Differential calculus and holomorphy" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Grothendieck,  "Résumé de la théorie métrique des produits tensoriels topologiques"  ''Bol. Soc. Mat. Sao-Paulo'' , '''8'''  (1956)  pp. 1–79</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,  "Topological vector spaces" , Gordon &amp; Breach  (1973)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Pisier,  "Factorization of linear operators and geometry of Banach spaces" , Amer. Math. Soc.  (1986)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Pisier,  "Counterexamples to a conjecture of Grothendieck"  ''Acta. Math.'' , '''151'''  (1983)  pp. 181–208</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.F. Colombeau,  "Differential calculus and holomorphy" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>

Revision as of 08:03, 6 June 2020


A locally convex space for which all continuous linear mappings into an arbitrary Banach space are nuclear operators (cf. Nuclear operator). The concept of a nuclear space arose [1] in an investigation of the question: For what spaces are the analogues of Schwartz' kernel theorem valid (see Nuclear bilinear form)? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [1]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear spaces play an important role in the spectral analysis of operators on Hilbert spaces (the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors, etc.) (see [2]). Nuclear spaces are closely connected with measure theory on locally convex spaces (see [3]). Nuclear spaces can be characterized in terms of dimension-type invariants (approximative dimension, diametral dimension, etc.) (see [2], [4], [5]). One of these invariants is the functional dimension, which for many spaces consisting of entire analytic functions is the same as the number of variables on which these functions depend (see [2]).

In their properties, nuclear spaces are close to finite-dimensional spaces. Every bounded set in a nuclear space is pre-compact. If a nuclear space is complete (or at least quasi-complete, that is, every closed bounded set is complete), then it is semi-reflexive (that is, the space coincides with its second dual as a set of elements), and every closed bounded set in it is compact. If a quasi-complete nuclear space is a barrelled space, then it is also a Montel space (in particular, a reflexive space); any weakly-convergent countable sequence in this space converges also in the original topology. A normed space is nuclear if and only if it is finite dimensional. Every nuclear space has the approximation property: Any continuous linear operator in such a space can be approximated in the operator topology of pre-compact convergence by operators of finite rank (that is, continuous linear operators with finite-dimensional ranges). Nevertheless, there are nuclear Fréchet spaces (cf. Fréchet space) that do not have the bounded approximation property; in such a space the identity operator is not the limit of a countable sequence of operators of finite rank in the strong or weak operator topology [6]. Nuclear Fréchet spaces without a Schauder basis have been constructed, and they can have arbitrarily small diametral dimension, that is, they can be arbitrarily near (in a certain sense) to finite-dimensional spaces [7]. For nuclear spaces a counterexample to the problem of invariant subspaces has been constructed: In a certain nuclear Fréchet space one can find a continuous linear operator without non-trivial invariant closed subspaces [8].

Examples of nuclear spaces.

1) Let $ {\mathcal E} ( \mathbf R ^ {n} ) $ be the space of all (real or complex) infinitely-differentiable functions on $ \mathbf R ^ {n} $ equipped with the topology of uniform convergence of all derivatives on compact subsets of $ \mathbf R ^ {n} $. The space $ {\mathcal E} ^ \prime ( \mathbf R ^ {n} ) $ dual to $ {\mathcal E} ( \mathbf R ^ {n} ) $ consists of all generalized functions (cf. Generalized function) with compact support. Let $ {\mathcal D} ( \mathbf R ^ {n} ) $ and $ {\mathcal S} ( \mathbf R ^ {n} ) $ be the linear subspaces of $ {\mathcal E} ( \mathbf R ^ {n} ) $ consisting, respectively, of functions with compact support and of functions that, together with all their derivatives, decrease faster than any power of $ | x | ^ {-} 1 $ as $ | x | \rightarrow \infty $. The duals $ {\mathcal D} ^ \prime ( \mathbf R ^ {n} ) $ and $ {\mathcal S} ^ \prime ( \mathbf R ^ {n} ) $ of $ {\mathcal D} ( \mathbf R ^ {n} ) $ and $ {\mathcal S} ( \mathbf R ^ {n} ) $, relative to the standard topology, consist of all generalized functions and of all generalized functions of slow growth, respectively. The spaces $ {\mathcal E} $, $ {\mathcal D} $, $ {\mathcal S} $, $ {\mathcal E} ^ \prime $, $ {\mathcal D} ^ \prime $, and $ {\mathcal S} ^ \prime $, equipped with the strong topology, are complete reflexive nuclear spaces.

2) let $ \{ a _ {np} \} $ be an infinite matrix, where $ 0 \leq a _ {np} < \infty $ and $ a _ {np} \leq a _ {n ( p + 1) } $, $ n, p = 1, 2 , . . . $. The space of sequences $ \xi = \{ \xi _ {n} \} $ for which $ | \xi | _ {p} = \sum _ {n = 0 } ^ \infty | \xi _ {n} | a _ {np} < \infty $ for all $ p $, with the topology defined by the semi-norms $ \xi \rightarrow | \xi | _ {p} $( cf. Semi-norm), is called a Köthe space, and is denoted by $ {\mathcal K} ( a _ {np} ) $. This space is nuclear if and only if for any $ p $ one can find a $ q $ such that $ \sum _ {n = 0 } ^ \infty ( a _ {np} /a _ {nq} ) < \infty $.

Heredity properties.

A locally convex space is nuclear if and only if its completion is nuclear. Every subspace (separable quotient space) of a nuclear space is nuclear. The direct sum, the inductive limit of a countable family of nuclear spaces, and also the product and the projective limit of any family of nuclear spaces, is again nuclear.

Let $ E $ be an arbitrary locally convex space, and let $ E ^ \prime $ denote its dual equipped with the strong topology. If $ E ^ \prime $ is nuclear, then $ E $ is called conuclear. If $ E $ is arbitrary and $ F $ is a nuclear space, then the space $ L ( E, F ) $ of continuous linear operators from $ E $ into $ F $ is nuclear with respect to the strong operator topology (simple convergence); if $ E $ is semi-reflexive and conuclear, then $ L ( E, F ) $ is nuclear also in the topology of bounded convergence.

Metric and dually-metric nuclear spaces.

A locally convex space $ E $ is called dually metric, or a space of type $ ( {\mathcal D} {\mathcal F} ) $, if it has a countable fundamental system of bounded sets and if every (strongly) bounded countable union of equicontinuous subsets in $ E ^ \prime $ is equicontinuous (cf. Equicontinuity). Any strong dual of a metrizable locally convex space is dually metric; the converse is not true. If $ E $ is a space of type $ ( {\mathcal D} {\mathcal F} ) $, then $ E ^ \prime $ is of type $ ( {\mathcal F} ) $( a Fréchet space, that is, complete and metrizable). Examples of nuclear spaces of type $ ( {\mathcal F} ) $ are Köthe spaces, and also $ {\mathcal E} $ and $ {\mathcal S} $; accordingly, $ {\mathcal E} ^ \prime $ and $ {\mathcal S} ^ \prime $ are nuclear spaces of type $ ( {\mathcal D} {\mathcal F} ) $. The spaces $ {\mathcal D} $ and $ {\mathcal D} ^ \prime $ are neither metric nor dually metric.

Metric and dually-metric nuclear spaces are separable, and if complete, they are reflexive. The transition $ E \rightarrow E ^ \prime $ to the dual space establishes a one-to-one correspondence between nuclear spaces of type $ ( {\mathcal F} ) $ and complete nuclear spaces of type $ ( {\mathcal D} {\mathcal F} ) $. If $ E $ is a complete nuclear space of type $ ( {\mathcal D} {\mathcal F} ) $ and if $ F $ is a nuclear space of type $ ( {\mathcal F} ) $, then $ L ( E, F ) $, equipped with the topology of bounded convergence, is nuclear and conuclear.

Every nuclear space of type $ ( {\mathcal F} ) $ is isomorphic to a subspace of the space $ {\mathcal E} ( \mathbf R ) $ of infinitely-differentiable functions on the real line, that is, $ {\mathcal E} ( \mathbf R ) $ is a universal space for the nuclear spaces of type $ ( {\mathcal F} ) $( see [10]). A Fréchet space $ E $ is nuclear if and only if every unconditionally-convergent series (cf. Unconditional convergence) in $ E $ is absolutely convergent (that is, with respect to any continuous semi-norm). Spaces of holomorphic functions on nuclear spaces of types $ ( {\mathcal F} ) $ and $ ( {\mathcal D} {\mathcal F} ) $ have been studied intensively (see [11]).

Tensor products of nuclear spaces, and spaces of vector functions.

The algebraic tensor product $ E \otimes F $ of two locally convex spaces $ E $ and $ F $ can be equipped with the projective and injective topologies, and then $ E \otimes F $ becomes a topological tensor product. The projective topology is the strongest locally convex topology in which the canonical bilinear mapping $ E \times F \rightarrow E \otimes F $ is continuous. The injective topology (or the topology of (bi) equicontinuous convergence) is induced by the natural imbedding $ E \otimes F \rightarrow L _ {e} ( E _ \tau ^ \prime , F ) $, where $ E _ \tau ^ \prime $ is the dual of $ E $ equipped with the Mackey topology $ \tau ( E ^ \prime , E) $, and $ L _ {e} ( E _ \tau ^ \prime , F ) $ is the space of continuous linear mappings $ E _ \tau ^ \prime \rightarrow F $ equipped with the topology of uniform convergence on equicontinuous sets in $ E ^ \prime $. Under this imbedding $ x \otimes y \in E \otimes F $ goes into the operator $ x ^ \prime \mapsto \langle x, x ^ \prime \rangle y $, where $ \langle x, x ^ \prime \rangle $ denotes the value of the functional $ x ^ \prime \in E ^ \prime $ at $ x \in E $. The completion of $ E \otimes F $ in the projective (respectively, injective) topology is denoted by $ E \widehat \otimes F $( respectively, $ E \widetilde \otimes F $).

For $ E $ to be a nuclear space it is necessary and sufficient that for any locally convex space $ F $ the projective and injective topologies in $ E \otimes F $ coincide, that is,

$$ \tag{1 } E \widehat \otimes F = E \widetilde \otimes F. $$

Actually, it suffices to require that (1) holds for $ F = l _ {1} $, the space of summable sequences, or for $ F $ equal to a fixed space with an unconditional basis (see [12]). Nevertheless, there is a (non-nuclear) infinite-dimensional separable Banach space $ X $ such that $ X \widehat \otimes X = X \widetilde \otimes X $( see [13]). If $ E $ and $ F $ are complete spaces and $ F $ is nuclear, then the imbedding $ E \otimes F \rightarrow L _ {e} ( E _ \tau ^ \prime , F ) $ can be extended to an isomorphism between $ E \widehat \otimes F $ and $ L _ {e} ( E _ \tau ^ \prime , F ) $.

If $ E $ is a non-null nuclear space, then $ E \widehat \otimes F $ is nuclear if and only if $ F $ is nuclear. If $ E $ and $ F $ are both spaces of type $ ( {\mathcal F} ) $( or $ ( {\mathcal D} {\mathcal F} ) $) and if $ E $ is nuclear, then $ ( E \widehat \otimes F ) ^ \prime = E ^ \prime \widehat \otimes F ^ \prime $.

Let $ E $ be a complete nuclear space consisting of scalar functions (not all) on a certain set $ T $; let also $ E $ be the inductive limit (locally convex hull) of a countable sequence of spaces of type $ ( {\mathcal F} ) $, and let the topology on $ E $ be not weaker than the topology of pointwise convergence of functions on $ T $. Then for any complete space $ F $ one can identify $ E \widehat \otimes F $ with the space of all mappings (vector functions) $ T \rightarrow F $ for which the scalar function $ t \mapsto \langle f ( t), y ^ \prime \rangle $ belongs to $ E $ for all $ y ^ \prime \in F ^ { \prime } $. In particular, $ {\mathcal E} ( \mathbf R ^ {n} ) \widehat \otimes F $ coincides with the space of all infinitely-differentiable vector functions on $ \mathbf R ^ {n} $ with values in $ F $, and $ {\mathcal E} ( \mathbf R ^ {n} ) \widehat \otimes {\mathcal E} ( \mathbf R ^ {m} ) = {\mathcal E} ( \mathbf R ^ {n} \times \mathbf R ^ {m} ) = {\mathcal E} ( \mathbf R ^ {n + m } ) $.

The structure of nuclear spaces.

Let $ U $ be a convex circled (i.e. convex balanced) neighbourhood of zero in a locally convex space $ E $, and let $ p $ be the Minkowski functional (continuous semi-norm) corresponding to $ U $. Let $ E _ {U} $ be the quotient space $ E/p ^ {-} 1 ( 0) $ with the norm induced by $ p $, and let $ {E _ {U} } hat $ be the completion of the normed space $ E _ {U} $. There is defined a continuous canonical linear mapping $ E \rightarrow {E _ {U} } hat $; if $ U $ contains a neighbourhood $ V $, then the continuous linear mapping $ {E _ {V} } hat \rightarrow {E _ {U} } hat $ is defined canonically.

For a locally convex space $ E $ the following conditions are equivalent: 1) $ E $ is nuclear; 2) $ E $ has a basis $ \mathfrak B $ of convex circled neighbourhoods of zero such that for any $ U \in \mathfrak B $ the canonical mapping $ E \rightarrow {E _ {U} } hat $ is a nuclear operator; 3) the mapping $ E \rightarrow {E _ {U} } hat $ is nuclear for any convex circled neighbourhood $ U $ of zero in $ E $; and 4) every convex circled neighbourhood $ U $ of zero in $ E $ contains another such neighbourhood of zero, $ V $, such that the canonical mapping $ {E _ {V} } hat \rightarrow {E _ {U} } hat $ is nuclear.

Let $ E $ be a nuclear space. For any neighbourhood $ U $ of zero in $ E $ and for any $ q $ such that $ 1 \leq q \leq \infty $ there is a convex circled neighbourhood $ V \subset U $ for which $ E _ {V} $ is (norm) isomorphic to a subspace of the space $ l _ {q} $ of sequences with summable $ q $- th powers. Thus, $ E $ coincides with a subspace of the projective limit of a family of spaces isomorphic to $ l _ {q} $. In particular (the case $ q = 2 $), in any nuclear space $ E $ there is a basis of neighbourhoods of zero $ \{ U _ \alpha \} $ such that all the spaces $ {E _ {U _ \alpha } } hat $ are Hilbert spaces; thus, $ E $ is Hilbertian, that is, the topology in $ E $ can be generated by a family of semi-norms each of which is obtained from a certain non-negative definite Hermitian form on $ E \times E $. Any complete nuclear space is isomorphic to the projective limit of a family of Hilbert spaces. A space $ E $ of type $ ( {\mathcal F} ) $ is nuclear if and only if it can be represented as the projective limit $ E = \lim\limits _ \leftarrow g _ {mn} H _ {n} $ of a countable family of Hilbert spaces $ H _ {n} $, such that the $ g _ {mn} $ are nuclear operators (or, at least, Hilbert–Schmidt operators, cf. Hilbert–Schmidt operator) for $ m < n $.

Bases in nuclear spaces.

In a nuclear space every equicontinuous basis is absolute. In a space of type $ ( {\mathcal F} ) $ any countable basis (even if weak) is an equicontinuous Schauder basis (cf. Basis), so that in a nuclear space of type $ ( {\mathcal F} ) $ any basis is absolute (in particular, unconditional). A similar result holds for complete nuclear spaces of type $ ( {\mathcal D} {\mathcal F} ) $, and for all nuclear spaces for which the closed-graph theorem holds. A quotient space of a nuclear space of type $ ( {\mathcal F} ) $ with a basis does not necessarily have a basis (see [4], [5], [6]).

Let $ E $ be a nuclear space of type $ ( {\mathcal F} ) $. A topology can be defined in $ E $ by a countable system of semi-norms $ x \mapsto \| x \| _ {q} $, $ q = 1, 2 \dots $ where $ \| x \| _ {q} \leq \| x \| _ {q + 1 } $ for all $ x \in E $. If $ E $ has a basis or a continuous norm, then the semi-norms $ \| \cdot \| $ can be taken as norms. Let $ \{ e _ {n} \} $ be a basis in $ E $; then any $ x \in E $ can be expressed as an (absolutely and unconditionally) convergent series

$$ x = \ \sum _ {n = 1 } ^ \infty \xi _ {n} e _ {n} , $$

where the coordinates $ \xi _ {n} $ have the form $ \xi _ {n} = \langle x, x _ {n} ^ \prime \rangle $, and the functionals $ x _ {n} ^ \prime $ form a bi-orthogonal basis in $ E ^ \prime $. $ E $ is isomorphic to the Köthe space $ {\mathcal K} ( a _ {nq} ) $, where $ a _ {nq} = \| e _ {n} \| _ {q} $; under this isomorphism $ x \in E $ goes into the sequence $ \{ \xi _ {n} \} $ of its coordinates. A basis $ \{ f _ {n} \} $ in $ E $ is equivalent to the basis $ \{ e _ {n} \} $( that is, it can be obtained from $ \{ e _ {n} \} $ by an isomorphism) if and only if $ {\mathcal K} ( \| e _ {n} \| _ {q} ) $ and $ {\mathcal K} ( \| f _ {n} \| _ {q} ) $ coincide as sets [4]. A basis $ \{ f _ {n} \} $ is called regular (or proper) if there is a system of norms $ \| \cdot \| _ {q} $ and a permutation $ \sigma $ of indices such that $ \| f _ {\sigma ( n) } \| _ {q} / \| f _ {\sigma ( n) } \| _ {r} $ is monotone decreasing for all $ r \geq q $. If a nuclear space $ E $ of type $ ( {\mathcal F} ) $ has a regular basis, then any two bases in $ E $ are quasi-equivalent (that is, they can be made equivalent by a permutation and a normalization of the elements of one of them). There are other sufficient conditions for all bases in $ E $ to be quasi-equivalent (see [4], [14]). A complete description of the class of nuclear spaces with this property is not known (1984).

Example. The Hermite functions $ \phi _ {n} ( t) = e ^ {t ^ {2} /2 } ( {d ^ {n} } / {dt ^ {n} } ) ( e ^ {- t ^ {2} } ) $ form a basis in the complete metric nuclear space $ {\mathcal S} ( \mathbf R ) $ of smooth functions on the real line that are rapidly decreasing together with all their derivatives. $ {\mathcal S} ( \mathbf R ) $ is isomorphic to $ {\mathcal K} ( n ^ {p} ) $.

References

[1] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[2] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)
[3] R.A. Minlos, "Generalized random processes and their extension in measure" Trudy Moskov. Mat. Obshch. , 8 (1959) pp. 497–518 (In Russian)
[4] B.S. Mityagin, "Approximate dimension and bases in nuclear spaces" Russian Math. Surveys , 16 : 4 pp. 59–127 Uspekhi Mat. Nauk , 16 : 4 (1961) pp. 63–132
[5] A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[6] E. Dubinsky, "Structure of nuclear Fréchet spaces" , Springer (1979)
[7] N.M. Zobin, B.S. Mityagin, "Examples of nuclear linear metric spaces without a basis" Funct. Anal. Appl. , 8 : 4 (1974) pp. 304–313 Funktsional. Anal. i Prilozhen. , 8 : 4 (1974) pp. 35–47
[8] A. Atzmon, "An operator without invariant subspaces on a nuclear Fréchet space" Ann. of Math. , 117 : 3 (1983) pp. 669–694
[9] H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[10] T. Komura, Y. Komura, "Ueber die Einbettung der nuklearen Räume in " Math. Ann. , 162 (1965–1966) pp. 284–288
[11] S. Dineen, "Complex analysis in locally convex spaces" , North-Holland (1981)
[12] K. John, V. Zizler, "On a tensor product characterization of nuclearity" Math. Ann. , 244 : 1 (1979) pp. 83–87
[13] G. Pisier, "Contre-example à une conjecture de Grothendieck" C.R. Acad. Sci. Paris , 293 (1981) pp. 681–683 (English abstract)
[14] M.M. Dragilev, "Bases in Köthe spaces" , Rostov-on-Don (1983) (In Russian)

Comments

A generalized function is also called a distribution, and a generalized function of slow growth is also called a tempered distribution.

Let $ F $ be a topological linear space, $ U $ a neighbourhood of zero in $ F $, $ A $ a set in $ F $, and $ \epsilon $ a (small) positive number. An $ \epsilon $- set for $ A $ relative to a neighbourhood $ U $ of zero is a set $ B $ such that for every $ a \in A $ there is a $ b \in B $ such that $ a \in b + \epsilon U $. Let $ N ( \epsilon , A , U ) $ be the smallest number of elements in $ \epsilon $- sets for $ A $ relative to $ U $. The functional dimension of $ F $ is defined by

$$ \mathop{\rm df} ( F ) = \sup _ { U } \inf _ { V } \ {\lim\limits \sup } _ {\epsilon \rightarrow 0 } \ \frac{ { \mathop{\rm ln} \mathop{\rm ln} } N ( \epsilon , V , U ) }{ { \mathop{\rm ln} \mathop{\rm ln} } \epsilon ^ {-} 1 } , $$

where $ U , V $ range over the neighbourhoods of zero in $ F $. Cf. [2], Sect. I.3.8 for more details.

Let $ F $ be a locally convex space and consider two neighbourhoods of zero $ U , V $ such that $ U $ absorbs $ V $, i.e. $ V \subset \rho U $ for some positive number $ \rho $. Let

$$ \delta _ {r} ( U , V ) = \inf \{ \delta : \exists \ \textrm{ subspace } G \textrm{ of dimension } \leq r $$

$$ {} \textrm{ such that } V \subset \delta U + G \} . $$

This number is called the $ r $- th diameter of $ V $ with respect to $ U $. The diametral dimension of a locally convex space is the collection of all sequences $ ( d _ {r} ) _ {r \in \mathbf N \cup \{ 0 \} } $ of non-negative numbers with the property that for each neighbourhood of zero $ U $ there is a neighbourhood $ V $ of zero absorbed by $ U $ for which $ \delta _ {r} ( U , V ) \leq d _ {r} $, $ r \in \mathbf N \cup \{ 0 \} $.

A locally convex space $ E $ is nuclear if and only if for some (respectively, each) positive number $ \lambda $ the sequence $ ( ( r + 1 ) ^ {- \lambda } ) _ {r \in N \cup \{ 0 \} } $ belongs to the diametral dimension of $ E $. See [5], Chapt. 9 for more details.

Let again $ U , V $ be neighbourhoods of zero of a locally convex space $ F $ such that $ U $ absorbs $ V $. The $ \epsilon $- content of $ V $ with respect to $ U $ is the supremum $ M _ \epsilon ( U , V ) $ of all natural numbers $ m $ such that there are $ x _ {1} \dots x _ {m} \in V $ with $ x _ {1} \dots x _ {k} \notin \epsilon U $ for all $ i \neq k $. The approximative dimension of a locally convex space $ F $ is the collection of all positive functions $ \phi $ on $ ( 0 , \infty ) $ such that for each neighbourhood $ U $ of zero there is a neighbourhood $ V $ of zero absorbed by $ U $ such that

$$ \lim\limits _ {\epsilon \rightarrow 0 } \phi ( \epsilon ) ^ {-} 1 M _ \epsilon ( U , V ) = 0 . $$

The number $ \rho ( U , V ) $ is defined by the exponential rate of growth of $ M _ \epsilon ( U , V ) $ as $ \epsilon \rightarrow 0 $. More precisely,

$$ \rho ( U , V ) = {\lim\limits \sup } _ {\epsilon \rightarrow 0 } \ \frac{ { \mathop{\rm ln} \mathop{\rm ln} } M _ \epsilon ( U , V ) }{ \mathop{\rm ln} \epsilon ^ {-} 1 } . $$

A locally convex space $ E $ is nuclear if and only if for some (respectively, each) positive number $ \rho $ the following condition is satisfied: For each neighbourhood of zero $ U $ there is a neighbourhood $ V $ of zero absorbed by $ U $ such that $ \rho ( U , V ) \leq \rho $. Cf. [5], Chapt. 9 for more details.

Let $ U $ be a bounded circled neighbourhood of a topological vector space $ F $. The Minkowski functional associated to $ U $ is defined by

$$ q ( x) = \inf _ {x \in \alpha U } \alpha ,\ \ \alpha \geq 0 . $$

This is well-defined for each $ x $ since $ U $ is absorbent (i.e. for each $ x \in F $ there is an $ \alpha $ such that $ x \in \alpha U $). Cf. [a7], Sects. 15.10, 16.4.

References

[a1] A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques" Bol. Soc. Mat. Sao-Paulo , 8 (1956) pp. 1–79
[a2] A. Grothendieck, "Topological vector spaces" , Gordon & Breach (1973) (Translated from French)
[a3] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
[a4] G. Pisier, "Factorization of linear operators and geometry of Banach spaces" , Amer. Math. Soc. (1986)
[a5] G. Pisier, "Counterexamples to a conjecture of Grothendieck" Acta. Math. , 151 (1983) pp. 181–208
[a6] J.F. Colombeau, "Differential calculus and holomorphy" , North-Holland (1982)
[a7] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
How to Cite This Entry:
Nuclear space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuclear_space&oldid=17938
This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article