In mathematics, nuclear operators are an important class of linear operators introduced by

Nuclear space at ncatlab

{{TopologicalTensorProductsAndNuclearSpaces Topological vector spaces Tensors

Alexander Grothendieck
Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative ...

in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times Y ...

of two topological vector spaces (TVSs).
Preliminaries and notation

Throughout let ''X'',''Y'', and ''Z'' be topological vector spaces (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated). * Theprojective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times Y ...

of two locally convex TVSs ''X'' and ''Y'' is denoted by $X\; \backslash otimes\_\; Y$ and the completion of this space will be denoted by $X\; \backslash widehat\_\; Y$.
* ''L'' : ''X'' → ''Y'' is a topological homomorphism or homomorphism, if it is linear, continuous, and $L\; :\; X\; \backslash to\; \backslash operatorname\; L$ is an open map, where $\backslash operatorname\; L$, the image of ''L'', has the subspace topology induced by ''Y''.
** If ''S'' is a subspace of ''X'' then both the quotient map ''X'' → ''X''/''S'' and the canonical injection ''S'' → ''X'' are homomorphisms.
* The set of continuous linear maps ''X'' → ''Z'' (resp. continuous bilinear maps $X\; \backslash times\; Y\; \backslash to\; Z$) will be denoted by L(''X'', ''Z'') (resp. B(''X'', ''Y''; ''Z'')) where if ''Z'' is the underlying scalar field then we may instead write L(''X'') (resp. B(''X'', ''Y'')).
* Any linear map $L\; :\; X\; \backslash to\; Y$ can be canonically decomposed as follows: $X\; \backslash to\; X\; /\; \backslash ker\; L\; \backslash ;\; \backslash xrightarrow\; \backslash ;\; \backslash operatorname\; L\; \backslash to\; Y$ where $L\_0\backslash left(\; x\; +\; \backslash ker\; L\; \backslash right)\; :=\; L\; (x)$ defines a bijection called the canonical bijection associated with ''L''.
* ''X''* or $X\text{'}$ will denote the continuous dual space of ''X''.
** To increase the clarity of the exposition, we use the common convention of writing elements of $X\text{'}$ with a prime following the symbol (e.g. $x\text{'}$ denotes an element of $X\text{'}$ and not, say, a derivative and the variables ''x'' and $x\text{'}$ need not be related in any way).
* $X^$ will denote the algebraic dual space of ''X'' (which is the vector space of all linear functionals on ''X'', whether continuous or not).
* A linear map ''L'' : ''H'' → ''H'' from a Hilbert space into itself is called positive if $\backslash langle\; L(x),\; X\; \backslash rangle\; \backslash geq\; 0$ for every $x\; \backslash in\; H$. In this case, there is a unique positive map ''r'' : ''H'' → ''H'', called the square-root of ''L'', such that $L\; =\; r\; \backslash circ\; r$.
** If $L\; :\; H\_1\; \backslash to\; H\_2$ is any continuous linear map between Hilbert spaces, then $L^*\; \backslash circ\; L$ is always positive. Now let ''R'' : ''H'' → ''H'' denote its positive square-root, which is called the absolute value of ''L''. Define $U\; :\; H\_1\; \backslash to\; H\_2$ first on $\backslash operatorname\; R$ by setting $U(x)\; =\; L(x)$ for $x\; =\; R\; \backslash left(\; x\_1\; \backslash right)\; \backslash in\; \backslash operatorname\; R$ and extending $U$ continuously to $\backslash overline$, and then define ''U'' on $\backslash ker\; R$ by setting $U(x)\; =\; 0$ for $x\; \backslash in\; \backslash ker\; R$ and extend this map linearly to all of $H\_1$. The map $U\backslash big\backslash vert\_\; :\; \backslash operatorname\; R\; \backslash to\; \backslash operatorname\; L$ is a surjective isometry and $L\; =\; U\; \backslash circ\; R$.
* A linear map $\backslash Lambda\; :\; X\; \backslash to\; Y$ is called compact or completely continuous if there is a neighborhood ''U'' of the origin in ''X'' such that $\backslash Lambda(U)$ is Totally bounded space, precompact in ''Y''.
** In a Hilbert space, positive compact linear operators, say ''L'' : ''H'' → ''H'' have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:

There is a sequence of positive numbers, decreasing and either finite or else converging to 0, $r\_1\; >\; r\_2\; >\; \backslash cdots\; >\; r\_k\; >\; \backslash cdots$ and a sequence of nonzero finite dimensional subspaces $V\_i$ of ''H'' (i = 1, 2, $\backslash ldots$) with the following properties: (1) the subspaces $V\_i$ are pairwise orthogonal; (2) for every ''i'' and every $x\; \backslash in\; V\_i$, $L(x)\; =\; r\_i\; x$; and (3) the orthogonal of the subspace spanned by $\backslash bigcup\_\; V\_i$ is equal to the kernel of ''L''.

Notation for topologies

* Topology of uniform convergence#The weak topology σ(X, X*), σ(X, X′) denotes the coarsest topology on ''X'' making every map in X′ continuous and $X\_$ or $X\_$ denotes Topology of uniform convergence#The weak topology σ(X, X*), ''X'' endowed with this topology. * Topology of uniform convergence#The weak topology σ(X*, X) or the weak* topology, σ(X′, X) denotes weak-* topology on X* and $X\_$ or $X\text{'}\_$ denotes Topology of uniform convergence#The weak topology σ(X′, X) or the weak* topology, X′ endowed with this topology. ** Note that every $x\_0\; \backslash in\; X$ induces a map $X\text{'}\; \backslash to\; \backslash mathbb$ defined by $\backslash lambda\; \backslash mapsto\; \backslash lambda\; \backslash left(\; x\_0\; \backslash right)$. ''σ''(X′, X) is the coarsest topology on X′ making all such maps continuous. * Topology of uniform convergence#Bounded convergence b(X, X*), b(X, X′) denotes the topology of bounded convergence on ''X'' and $X\_$ or $X\_$ denotes Topology of uniform convergence#Bounded convergence b(X, X*), ''X'' endowed with this topology. * Topology of uniform convergence#Strong dual topology b(X*, X), b(X′, X) denotes the topology of bounded convergence on X′ or the strong dual topology on X′ and $X\_$ or $X\text{'}\_$ denotes Topology of uniform convergence#Strong dual topology b(X*, X), X′ endowed with this topology. ** As usual, if X* is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be b(X′, X).A canonical tensor product as a subspace of the dual of Bi(X, Y)

Let ''X'' and ''Y'' be vector spaces (no topology is needed yet) and let Bi(''X'', ''Y'') be the space of all bilinear maps defined on $X\; \backslash times\; Y$ and going into the underlying scalar field. For every $(x,\; y)\; \backslash in\; X\; \backslash times\; Y$, let $\backslash chi\_$ be the canonical bilinear form on Bi(''X'', ''Y'') defined by $\backslash chi\_(u)\; :=\; u(x,\; y)$ for every ''u'' ∈ Bi(''X'', ''Y''). This induces a canonical map $\backslash chi\; :\; X\; \backslash times\; Y\; \backslash to\; \backslash mathrm(X,\; Y)^$ defined by $\backslash chi(x,\; y)\; :=\; \backslash chi\_$, where $\backslash mathrm(X,\; Y)^$ denotes the algebraic dual of Bi(''X'', ''Y''). If we denote the span of the range of ''𝜒'' by ''X'' ⊗ ''Y'' then it can be shown that ''X'' ⊗ ''Y'' together with ''𝜒'' forms a tensor product of ''X'' and ''Y'' (where ''x'' ⊗ ''y := ''𝜒''(''x'', ''y'')). This gives us a canonical tensor product of ''X'' and ''Y''. If ''Z'' is any other vector space then the mapping Li(''X'' ⊗ ''Y''; ''Z'') → Bi(''X'', ''Y''; ''Z'') given by ''u'' ↦ ''u'' ∘ ''𝜒'' is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of ''X'' ⊗ ''Y'' with the space of bilinear forms on ''X'' × ''Y''. Moreover, if ''X'' and ''Y'' are locally convex topological vector spaces (TVSs) and if ''X'' ⊗ ''Y'' is given the 𝜋-topology then for every locally convex TVS ''Z'', this map restricts to a vector space isomorphism $L(X\; \backslash otimes\_\; Y;\; Z)\; \backslash to\; B(X,\; Y;\; Z)$ from the space of ''continuous'' linear mappings onto the space of ''continuous'' bilinear mappings. In particular, the continuous dual of ''X'' ⊗ ''Y'' can be canonically identified with the space B(''X'', ''Y'') of continuous bilinear forms on ''X'' × ''Y''; furthermore, under this identification the equicontinuous subsets of B(''X'', ''Y'') are the same as the equicontinuous subsets of ''$(X\; \backslash otimes\_\; Y)\text{'}$.Nuclear operators between Banach spaces

There is a canonical vector space embedding $I\; :\; X\text{'}\; \backslash otimes\; Y\; \backslash to\; L(X;\; Y)$ defined by sending $z\; :=\; \backslash sum\_^n\; x\_i\text{'}\; \backslash otimes\; y\_i$ to the map : $x\; \backslash mapsto\; \backslash sum\_^n\; x\_i\text{'}(x)\; y\_i\; .$ Assuming that ''X'' and ''Y'' are Banach spaces, then the map $I\; :\; X\text{'}\_b\; \backslash otimes\_\; Y\; \backslash to\; L\_b(X;\; Y)$ has norm $1$ (to see that the norm is $\backslash leq\; 1$, note that $\backslash ,\; I(z)\; \backslash ,\; =\; \backslash sup\_\; \backslash ,\; I(z)(x)\; \backslash ,\; =\; \backslash sup\_\; \backslash left\backslash ,\; \backslash sum\_^\; x\_i\text{'}(x)\; y\_i\; \backslash right\backslash ,\; \backslash leq\; \backslash sup\_\; \backslash sum\_^\; \backslash left\backslash ,\; x\_i\text{'}\; \backslash right\backslash ,\; \backslash ,\; x\backslash ,\; \backslash left\backslash ,\; y\_i\; \backslash right\backslash ,\; \backslash leq\; \backslash sum\_^\; \backslash left\backslash ,\; x\_i\text{'}\; \backslash right\backslash ,\; \backslash left\backslash ,\; y\_i\; \backslash right\backslash ,$ so that $\backslash left\backslash ,\; I(z)\; \backslash right\backslash ,\; \backslash leq\; \backslash left\backslash ,\; z\; \backslash right\backslash ,\; \_$). Thus it has a continuous extension to a map $\backslash hat\; :\; X\text{'}\_b\; \backslash widehat\_\; Y\; \backslash to\; L\_b(X;\; Y)$, where it is known that this map is not necessarily injective. The range of this map is denoted by $L^1(X;\; Y)$ and its elements are called nuclear operators. $L^1(X;\; Y)$ is TVS-isomorphic to $\backslash left(\; X\text{'}\_b\; \backslash widehat\_\; Y\; \backslash right)\; /\; \backslash ker\; \backslash hat$ and the norm on this quotient space, when transferred to elements of $L^1(X;\; Y)$ via the induced map $\backslash hat\; :\; \backslash left(\; X\text{'}\_b\; \backslash widehat\_\; Y\; \backslash right)\; /\; \backslash ker\; \backslash hat\; \backslash to\; L^1(X;\; Y)$, is called the trace-norm and is denoted by $\backslash ,\; \backslash cdot\; \backslash ,\; \_$. Explicitely, if $T\; :\; X\; \backslash to\; Y$ is a nuclear operator then $\backslash left\backslash ,\; T\; \backslash right\backslash ,\; \_\; :=\; \backslash inf\_\; \backslash left\backslash ,\; z\; \backslash right\backslash ,\; \_$.Characterization

Suppose that ''X'' and ''Y'' are Banach spaces and that $N\; :\; X\; \backslash to\; Y$ is a continuous linear operator. * The following are equivalent: *# $N\; :\; X\; \backslash to\; Y$ is nuclear. *# There exists an sequence $\backslash left(\; x\_i\text{'}\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in the closed unit ball of $X\text{'}$, a sequence $\backslash left(\; y\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in the closed unit ball of $Y$, and a complex sequence $\backslash left(\; c\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ such that $\backslash sum\_\{i=1\}^\{\backslash infty\}\; ,\; c\_i,\; <\; \backslash infty$ and $N$ is equal to the mapping: $N(x)\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; c\_i\; x\text{'}\_i(x)\; y\_i$ for all $x\; \backslash in\; X$. Furthermore, the trace-norm $\backslash ,\; N\; \backslash ,\; \_\{\backslash operatorname\{Tr$ is equal to the infimum of the numbers $\backslash sum\_\{i=1\}^\{\backslash infty\}\; ,\; c\_i\; ,$ over the set of all representations of $N$ as such a series. * If ''Y'' is Reflexive space, reflexive then $N\; :\; X\; \backslash to\; Y$ is a nuclear if and only if $\{\}^\{t\}N\; :\; Y\text{'}\_\{b\}\; \backslash to\; X\text{'}\_\{b\}$ is nuclear, in which case $\backslash left\backslash ,\; \{\}^\{t\}N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; =\; \backslash left\backslash ,\; N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr$.Properties

Let ''X'' and ''Y'' be Banach spaces and let $N\; :\; X\; \backslash to\; Y$ be a continuous linear operator. * If $N\; :\; X\; \backslash to\; Y$ is a nuclear map then its transpose $\{\}^\{t\}N\; :\; Y\text{'}\_\{b\}\; \backslash to\; X\text{'}\_\{b\}$ is a continuous nuclear map (when the dual spaces carry their strong dual topologies) and $\backslash left\backslash ,\; \{\}^\{t\}N\backslash right\; \backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash leq\; \backslash left\backslash ,\; N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr$.Nuclear operators between Hilbert spaces

Nuclear automorphisms of a Hilbert space are called trace class operators. Let ''X'' and ''Y'' be Hilbert spaces and let ''N'' : ''X'' → ''Y'' be a continuous linear map. Suppose that $N\; =\; U\; \backslash circ\; R$ where ''R'' : ''X'' → ''X'' is the square-root of $N^*\; \backslash circ\; N$ and ''U'' : ''X'' → ''Y'' is such that $U\backslash big\backslash vert\_\{\backslash operatorname\{Im\}\; R\}\; :\; \backslash operatorname\{Im\}\; R\; \backslash to\; \backslash operatorname\{Im\}\; L$ is a surjective isometry and $N\; =\; U\; \backslash circ\; R$. Then ''N'' is a nuclear map if and only if ''R'' is a nuclear map; hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive linear operators.Characterizations

Let ''X'' and ''Y'' be Hilbert spaces and let ''N'' : ''X'' → ''Y'' be a continuous linear map whose absolute value is ''R'' : ''X'' → ''X''. The following are equivalent: #''N'' : ''X'' → ''Y'' is nuclear. #''R'' : ''X'' → ''X'' is nuclear. #''R'' : ''X'' → ''X'' is compact and $\backslash operatorname\{Tr\}\; R$ is finite, in which case $\backslash operatorname\{Tr\}\; R\; =\; \backslash ,\; N\; \backslash ,\; \_\{\backslash operatorname\{Tr$. #* Here, $\backslash operatorname\{Tr\}\; R$ is the trace of ''R'' and it is defined as follows: Since ''R'' is a continuous compact positive operator, there exists a (possibly finite) sequence $\backslash lambda\_1\; >\; \backslash lambda\_2\; >\; \backslash cdots$ of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces $V\_1,\; V\_2,\; \backslash ldots$ such that the orthogonal (in ''H'') of $\backslash operatorname\{span\}\backslash left(\; V\_1\; \backslash cup\; V\_2\; \backslash cup\; \backslash cdots\; \backslash right)$ is equal to $\backslash ker\; R$ (and hence also to $\backslash ker\; N$) and for all ''k'', $R(x)\; =\; \backslash lambda\_k\; x$ for all $x\; \backslash in\; V\_k$; the trace is defined as $\backslash operatorname\{Tr\}\; R\; :=\; \backslash sum\_\{k\}\; \backslash lambda\_k\; \backslash dim\; V\_k$. #$\{\}^\{t\}N\; :\; Y\text{'}\_\{b\}\; \backslash to\; X\text{'}\_\{b\}$ is nuclear, in which case $\backslash ,\; \{\}^\{t\}N\; \backslash ,\; \_\{\backslash operatorname\{Tr\; =\; \backslash ,\; N\; \backslash ,\; \_\{\backslash operatorname\{Tr$. #There are two orthogonal sequences $\backslash left(\; x\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in ''X'' and $\backslash left(\; y\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in ''Y'', and a sequence $\backslash left(\; \backslash lambda\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in $l^1$ such that for all $x\; \backslash in\; X$, $N(x)\; =\; \backslash sum\_\{i\}\; \backslash lambda\_i\; \backslash langle\; x,\; x\_i\; \backslash rangle\; y\_i$. #''N'' : ''X'' → ''Y' is an integral map.Nuclear operators between locally convex spaces

Suppose that ''U'' is a convex balanced closed neighborhood of the origin in ''X'' and ''B'' is a convex balanced bounded Banach disk in ''Y'' with both ''X'' and ''Y'' locally convex spaces. Let $p\_U(x)\; =\; \backslash inf\_\{r\; >\; 0,\; x\; \backslash in\; r\; U\}\; r$ and let $\backslash pi\; :\; X\; \backslash to\; X/p\_U^\{-1\}(0)$ be the canonical projection. One can define the Auxiliary normed spaces, auxiliary Banach space $\backslash hat\{X\}\_U$ with the canonical map $\backslash hat\{\backslash pi\}\_U\; :\; X\; \backslash to\; \backslash hat\{X\}\_U$ whose image, $X/p\_U^\{-1\}(0)$, is dense in $\backslash hat\{X\}\_U$ as well as the auxiliary space $F\_B\; =\; \backslash operatorname\{span\}\; B$ normed by $p\_B(y)\; =\; \backslash inf\_\{r\; >\; 0,\; y\; \backslash in\; r\; B\}\; r$ and with a canonical map $\backslash iota\; :\; F\_B\; \backslash to\; F$ being the (continuous) canonical injection. Given any continuous linear map $T\; :\; \backslash hat\{X\}\_U\; \backslash to\; Y\_B$ one obtains through composition the continuous linear map $\backslash hat\{\backslash pi\}\_U\; \backslash circ\; T\; \backslash circ\; \backslash iota\; :\; X\; \backslash to\; Y$; thus we have an injection $L\; \backslash left(\; \backslash hat\{X\}\_U;\; Y\_B\; \backslash right)\; \backslash to\; L(X;\; Y)$ and we henceforth use this map to identify $L\; \backslash left(\; \backslash hat\{X\}\_U;\; Y\_B\; \backslash right)$ as a subspace of $L(X;\; Y)$. Definition: Let ''X'' and ''Y'' be Hausdorff locally convex spaces. The union of all $L^1\backslash left(\; \backslash hat\{X\}\_U;\; Y\_B\; \backslash right)$ as ''U'' ranges over all closed convex balanced neighborhoods of the origin in ''X'' and ''B'' ranges over all bounded Banach disks in ''Y'', is denoted by $L^1(X;\; Y)$ and its elements are call nuclear mappings of ''X'' into ''Y''. When ''X'' and ''Y'' are Banach spaces, then this new definition of ''nuclear mapping'' is consistent with the original one given for the special case where ''X'' and ''Y'' are Banach spaces.Sufficient conditions for nuclearity

* Let ''W'', ''X'', ''Y'', and ''Z'' be Hausdorff locally convex spaces, $N\; :\; X\; \backslash to\; Y$ a nuclear map, and $M\; :\; W\; \backslash to\; X$ and $P\; :\; Y\; \backslash to\; Z$ be continuous linear maps. Then $N\; \backslash circ\; M\; :\; W\; \backslash to\; Y$, $P\; \backslash circ\; N\; :\; X\; \backslash to\; Z$, and $P\; \backslash circ\; N\; \backslash circ\; M\; :\; W\; \backslash to\; Z$ are nuclear and if in addition ''W'', ''X'', ''Y'', and ''Z'' are all Banach spaces then $\backslash left\backslash ,\; P\; \backslash circ\; N\; \backslash circ\; M\backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash leq\; \backslash left\backslash ,\; P\; \backslash right\backslash ,\; \backslash left\backslash ,\; N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash ,\; \backslash left\backslash ,\; M\; \backslash right\backslash ,$. * If $N\; :\; X\; \backslash to\; Y$ is a nuclear map between two Hausdorff locally convex spaces, then its transpose $\{\}^\{t\}N\; :\; Y\text{'}\_\{b\}\; \backslash to\; X\text{'}\_\{b\}$ is a continuous nuclear map (when the dual spaces carry their strong dual topologies). ** If in addition ''X'' and ''Y'' are Banach spaces, then $\backslash left\backslash ,\; \{\}^\{t\}N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash leq\; \backslash left\backslash ,\; N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr$. * If $N\; :\; X\; \backslash to\; Y$ is a nuclear map between two Hausdorff locally convex spaces and if $\backslash hat\{X\}$ is a completion of ''X'', then the unique continuous extension $\backslash hat\{N\}\; :\; \backslash hat\{X\}\; \backslash to\; Y$ of ''N'' is nuclear.Characterizations

Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let $N\; :\; X\; \backslash to\; Y$ be a continuous linear operator. * The following are equivalent: *# $N\; :\; X\; \backslash to\; Y$ is nuclear. *# (Definition) There exists a convex balanced neighborhood ''U'' of the origin in ''X'' and a bounded Banach disk ''B'' in ''Y'' such that $N(U)\; \backslash subseteq\; B$ and the induced map $\backslash overline\{N\}\_0\; :\; \backslash hat\{X\}\_U\; \backslash to\; Y\_B$ is nuclear, where $\backslash overline\{N\}\_0$ is the unique continuous extension of $N\_0\; :\; X\_U\; \backslash to\; Y\_B$, which is the unique map satisfying $N\; =\; \backslash operatorname\{In\}\_B\; \backslash circ\; N\_0\; \backslash circ\; \backslash pi\_U$ where $\backslash operatorname\{In\}\_B\; :\; Y\_B\; \backslash to\; Y$ is the natural inclusion and $\backslash pi\_U\; :\; X\; \backslash to\; X\; /\; p\_U^\{-1\}(0)$ is the canonical projection. *# There exist Banach spaces $B\_1$ and $B\_2$ and continuous linear maps $f\; :\; X\; \backslash to\; B\_1$, $n\; :\; B\_1\; \backslash to\; B\_2$, and $g\; :\; B\_2\; \backslash to\; Y$ such that $n\; :\; B\_1\; \backslash to\; B\_2$ is nuclear and $N\; =\; g\; \backslash circ\; n\; \backslash circ\; f$. *# There exists an equicontinuous sequence $\backslash left(\; x\_i\text{'}\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X\text{'}$, a bounded Banach disk $B\; \backslash subseteq\; Y$, a sequence $\backslash left(\; y\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in ''B'', and a complex sequence $\backslash left(\; c\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ such that $\backslash sum\_\{i=1\}^\{\backslash infty\}\; ,\; c\_i,\; <\; \backslash infty$ and $N$ is equal to the mapping: $N(x)\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; c\_i\; x\text{'}\_i(x)\; y\_i$ for all $x\; \backslash in\; X$. * If ''X'' is barreled and ''Y'' is quasi-complete, then ''N'' is nuclear if and only if ''N'' has a representation of the form $N(x)\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; c\_i\; x\text{'}\_i(x)\; y\_i$ with $\backslash left(\; x\_i\text{'}\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ bounded in $X\text{'}$, $\backslash left(\; y\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ bounded in ''Y'' and $\backslash sum\_\{i=1\}^\{\backslash infty\}\; ,\; c\_i,\; <\; \backslash infty$.Properties

The following is a type of ''Hahn-Banach theorem'' for extending nuclear maps: * If $E\; :\; X\; \backslash to\; Z$ is a TVS-embedding and $N\; :\; X\; \backslash to\; Y$ is a nuclear map then there exists a nuclear map $\backslash tilde\{N\}\; :\; Z\; \backslash to\; Y$ such that $\backslash tilde\{N\}\; \backslash circ\; E\; =\; N$. Furthermore, when ''X'' and ''Y'' are Banach spaces and ''E'' is an isometry then for any $\backslash epsilon\; >\; 0$, $\backslash tilde\{N\}$ can be picked so that $\backslash ,\; \backslash tilde\{N\}\; \backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash leq\; \backslash ,\; N\; \backslash ,\; \_\{\backslash operatorname\{Tr\; +\; \backslash epsilon$. * Suppose that $E\; :\; X\; \backslash to\; Z$ is a TVS-embedding whose image is closed in ''Z'' and let $\backslash pi\; :\; Z\; \backslash to\; Z\; /\; \backslash operatorname\{Im\}\; E$ be the canonical projection. Suppose all that every compact disk in $Z\; /\; \backslash operatorname\{Im\}\; E$ is the image under $\backslash pi$ of a bounded Banach disk in ''Z'' (this is true, for instance, if ''X'' and ''Z'' are both Fréchet spaces, or if ''Z'' is the strong dual of a Fréchet space and $\backslash operatorname\{Im\}\; E$ is weakly closed in ''Z''). Then for every nuclear map $N\; :\; Y\; \backslash to\; Z\; /\; \backslash operatorname\{Im\}\; E$ there exists a nuclear map $\backslash tilde\{N\}\; :\; Y\; \backslash to\; Z$ such that $\backslash pi\; \backslash circ\; \backslash tilde\{N\}\; =\; N$. ** Furthermore, when ''X'' and ''Z'' are Banach spaces and ''E'' is an isometry then for any $\backslash epsilon\; >\; 0$, $\backslash tilde\{N\}$ can be picked so that $\backslash left\backslash ,\; \backslash tilde\{N\}\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; \backslash leq\; \backslash left\backslash ,\; N\; \backslash right\backslash ,\; \_\{\backslash operatorname\{Tr\; +\; \backslash epsilon$. Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let $N\; :\; X\; \backslash to\; Y$ be a continuous linear operator. * Any nuclear map is compact. * For every topology of uniform convergence on $L(\; X;\; Y\; )$, the nuclear maps are contained in the closure of $X\text{'}\; \backslash otimes\; Y$ (when $X\text{'}\; \backslash otimes\; Y$ is viewed as a subspace of $L(\; X;\; Y\; )$).See also

* * * * * * * * * * * *References

Bibliography

* * * * * * * * * * * * * *External links

Nuclear space at ncatlab

{{TopologicalTensorProductsAndNuclearSpaces Topological vector spaces Tensors