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In mathematics, '''nuclear operators''' are an important class of linear operators introduced by [[Biography:Alexander Grothendieck|Alexander Grothendieck]] in his doctoral dissertation. Nuclear operators are intimately tied to the [[Projective tensor product|projective tensor product]] of two [[Topological vector space|topological vector space]]s (TVSs).

== Preliminaries and notation ==

Throughout let ''X'',''Y'', and ''Z'' be [[Topological vector space|topological vector space]]s (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated).

* The [[Projective tensor product|projective tensor product]] of two locally convex TVSs ''X'' and ''Y'' is denoted by <math>X \otimes_{\pi} Y</math> and the completion of this space will be denoted by <math>X \widehat{\otimes}_{\pi} Y</math>. 
* ''L'' : ''X'' → ''Y'' is a '''[[Topological homomorphism|topological homomorphism]]''' or '''homomorphism''', if it is linear, continuous, and <math>L : X \to \operatorname{Im} L</math> is an open map, where <math>\operatorname{Im} L</math>, 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 <math>X \times Y \to Z</math>) 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 <math>L : X \to Y</math> can be canonically decomposed as follows: <math>X \to X / \ker L \; \xrightarrow{L_0} \; \operatorname{Im} L \to Y</math> where <math>L_0\left( x + \ker L \right) := L (x)</math> defines a bijection called the '''canonical bijection''' associated with ''L''. 
* ''X''* or <math>X'</math> will denote the continuous dual space of ''X''.
** To increase the clarity of the exposition, we use the common convention of writing elements of <math>X'</math> with a prime following the symbol (e.g. <math>x'</math> denotes an element of <math>X'</math> and not, say, a derivative and the variables ''x'' and <math>x'</math> need not be related in any way). 
* <math>X^{\#}</math> 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 <math>\langle L(x), x  \rangle \geq 0</math> for every <math>x \in H</math>. In this case, there is a unique positive map ''r'' : ''H'' → ''H'', called the '''square-root''' of ''L'', such that <math>L = r \circ r</math>.{{sfn | Trèves | 2006 | p=488}} 
** If <math>L : H_1 \to H_2</math> is any continuous linear map between Hilbert spaces, then <math>L^* \circ L</math> is always positive. Now let ''R'' : ''H'' → ''H'' denote its positive square-root, which is called the '''absolute value''' of ''L''. Define <math>U : H_1 \to H_2</math> first on <math>\operatorname{Im} R</math> by setting <math>U(x) = L(x)</math> for <math>x = R \left( x_1 \right) \in \operatorname{Im} R</math> and extending <math>U</math> continuously to <math>\overline{\operatorname{Im} R}</math>, and then define ''U'' on <math>\ker R</math> by setting <math>U(x) = 0</math> for <math>x \in \ker R</math> and extend this map linearly to all of <math>H_1</math>. The map <math>U\big\vert_{\operatorname{Im} R} : \operatorname{Im} R \to \operatorname{Im} L</math> is a surjective isometry and <math>L = U \circ R</math>. 
* A linear map <math>\Lambda : X \to Y</math> is called '''compact''' or '''completely continuous''' if there is a neighborhood ''U'' of the origin in ''X'' such that <math>\Lambda(U)</math> is [[Totally bounded space|precompact]] in ''Y''.{{sfn | Trèves | 2006 | p=483}}
** <p>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:{{sfn | Trèves | 2006 | p=490}}</p><p>There is a sequence of positive numbers, decreasing and either finite or else converging to 0, <math>r_1 > r_2 > \cdots > r_k > \cdots</math> and a sequence of nonzero finite dimensional subspaces <math>V_i</math> of ''H'' (i = 1, 2, <math>\ldots</math>) with the following properties: (1) the subspaces <math>V_i</math> are pairwise orthogonal; (2) for every ''i'' and every <math>x \in V_i</math>, <math>L(x) = r_i x</math>; and (3) the orthogonal of the subspace spanned by <math display="inline">\bigcup_{i} V_i</math> is equal to the kernel of ''L''.{{sfn | Trèves | 2006 | p=490}}</p>

=== Notation for topologies ===
{{Main|Topology of uniform convergence|Mackey topology}}
* [[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 <math>X_{\sigma\left(X, X'\right)}</math> or  <math>X_{\sigma}</math> 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 <math>X_{\sigma\left(X', X\right)}</math> or  <math>X'_{\sigma}</math> denotes [[Topology of uniform convergence#The weak topology σ(X′, X) or the weak* topology|X′ endowed with this topology]]. 
** Note that every <math>x_0 \in X</math> induces a map <math>X' \to \mathbb{R}</math> defined by <math>\lambda \mapsto \lambda \left( x_0 \right)</math>. ''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 <math>X_{b\left(X, X'\right)}</math> or <math>X_{b}</math> 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 <math>X_{b\left(X', X\right)}</math> or <math>X'_{b}</math> 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 map|bilinear map]]s defined on <math>X \times Y</math> and going into the underlying scalar field.

For every <math>(x, y) \in X \times Y</math>, let <math>\chi_{(x, y)}</math> be the canonical linear form on Bi(''X'', ''Y'') defined by <math>\chi_{(x, y)}(u) := u(x, y)</math> for every ''u'' ∈ Bi(''X'', ''Y''). 
This induces a canonical map <math>\chi : X \times Y \to \mathrm{Bi}(X, Y)^{\#}</math> defined by <math>\chi(x, y) := \chi_{(x, y)}</math>, where <math>\mathrm{Bi}(X, Y)^{\#}</math> 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|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''.{{sfn | Schaefer|Wolff| 1999 | p=92}} 
Moreover, if ''X'' and ''Y'' are locally convex [[Topological vector space|topological vector space]]s (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 <math>L(X \otimes_{\pi} Y; Z) \to B(X, Y; Z)</math> from the space of ''continuous'' linear mappings onto the space of ''continuous'' bilinear mappings.{{sfn | Schaefer|Wolff| 1999 | p=93}} 
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 <math>(X \otimes_{\pi} Y)'</math>.{{sfn | Schaefer|Wolff| 1999 | p=93}}

== Nuclear operators between Banach spaces ==
{{Main|Nuclear operators between Banach spaces|Projective tensor product}}
There is a canonical vector space embedding <math>I : X' \otimes Y \to L(X; Y)</math> defined by sending <math display="inline">z := \sum_{i}^n x_i' \otimes y_i</math> to the map
: <math>x \mapsto \sum_{i}^n x_i'(x) y_i .</math>
Assuming that ''X'' and ''Y'' are Banach spaces, then the map <math>I : X'_b \otimes_{\pi} Y \to L_b(X; Y)</math> has norm <math>1</math> (to see that the norm is <math>\leq 1</math>, note that <math display="inline">\| I(z) \| = \sup_{\| x \| \leq 1} \| I(z)(x) \| = \sup_{\| x \| \leq 1} \left\| \sum_{i=1}^{n} x_i'(x) y_i \right\| \leq \sup_{\| x \| \leq 1} \sum_{i=1}^{n} \left\| x_i' \right\| \|x\| \left\| y_i \right\| \leq \sum_{i=1}^{n} \left\| x_i' \right\| \left\| y_i \right\|</math> so that <math>\left\| I(z) \right\| \leq \left\| z \right\|_{\pi}</math>). Thus it has a continuous extension to a map <math>\hat{I} : X'_b \widehat{\otimes}_{\pi} Y \to L_b(X; Y)</math>, where it is known that this map is not necessarily injective.{{sfn | Schaefer|Wolff| 1999 | p=98}} The range of this map is denoted by <math>L^1(X; Y)</math> and its elements are called '''nuclear operators'''.{{sfn | Trèves | 2006 | pp=478-479}} <math>L^1(X; Y)</math> is TVS-isomorphic to <math>\left( X'_b \widehat{\otimes}_{\pi} Y \right) / \ker \hat{I}</math> and the norm on this quotient space, when transferred to elements of <math>L^1(X; Y)</math> via the induced map <math>\hat{I} : \left( X'_b \widehat{\otimes}_{\pi} Y \right) / \ker \hat{I} \to L^1(X; Y)</math>, is called the '''trace-norm''' and is denoted by <math>\| \cdot \|_{\operatorname{Tr}}</math>. Explicitly,{{clarify|post-text=explicitly or especially?|date=January 2021}} if <math>T : X \to Y</math> is a nuclear operator then <math display="inline">\left\| T \right\|_{\operatorname{Tr}} := \inf_{z \in \hat{I}^{-1}\left( T \right)} \left\| z \right\|_{\pi} </math>.
<!-- \left\| T \right\|_{\operatorname{Tr} 
= \inf \left\{ \left\| \sum_{i=1}^{\infty} \lambda_i x_i' \otimes y_i \right\|_{\pi} : \hat{I}\left( \sum_{i=1}^{\infty} \lambda_i x_i' \otimes y_i \right) = T, \sum_{i=1}^{\infty} | \lambda_i | < \infty, \left( x_i' \right)_{i=1}^{\infty} \subseteq X' \text{ and } \left( y_i \right)_{i=1}^{\infty} \subseteq Y \text{ are null sequences} \right\}-->

=== Characterization ===
Suppose that ''X'' and ''Y'' are Banach spaces and that <math>N : X \to Y</math> is a continuous linear operator. 
* The following are equivalent:
*# <math>N : X \to Y</math> is nuclear.
*# There exists an sequence <math>\left( x_i' \right)_{i=1}^{\infty}</math> in the closed unit ball of <math>X'</math>, a sequence <math>\left( y_i \right)_{i=1}^{\infty}</math> in the closed unit ball of <math>Y</math>, and a complex sequence <math>\left( c_i \right)_{i=1}^{\infty}</math> such that <math display="inline">\sum_{i=1}^{\infty} |c_i| < \infty</math> and <math>N</math> is equal to the mapping:{{sfn | Trèves | 2006 | pp=481-483}} <math display="inline">N(x) = \sum_{i=1}^{\infty} c_i x'_i(x) y_i</math> for all <math>x \in X</math>. Furthermore, the trace-norm <math>\| N \|_{\operatorname{Tr}}</math> is equal to the infimum of the numbers <math display="inline">\sum_{i=1}^{\infty} | c_i |</math> over the set of all representations of <math>N</math> as such a series.{{sfn | Trèves | 2006 | pp=481-483}}
* If ''Y'' is [[Reflexive space|reflexive]] then <math>N : X \to Y</math> is a nuclear if and only if <math>{}^{t}N : Y'_{b} \to X'_{b}</math> is nuclear, in which case <math display="inline">\left\| {}^{t}N \right\|_{\operatorname{Tr}} = \left\| N \right\|_{\operatorname{Tr}}</math>. {{sfn | Trèves | 2006 | p=484}}

=== Properties ===
Let ''X'' and ''Y'' be Banach spaces and let <math>N : X \to Y</math> be a continuous linear operator. 
* If <math>N : X \to Y</math> is a nuclear map then its transpose <math>{}^{t}N : Y'_{b} \to X'_{b}</math> is a continuous nuclear map (when the dual spaces carry their strong dual topologies) and <math display="inline">\left\| {}^{t}N\right \|_{\operatorname{Tr}} \leq \left\| N \right\|_{\operatorname{Tr}}</math>.{{sfn | Trèves | 2006 | pp=483-484}}

== Nuclear operators between Hilbert spaces ==


Nuclear [[Automorphism|automorphism]]s of a [[Hilbert space]] are called '''[[Trace class|trace class]]''' operators.

Let ''X'' and ''Y'' be Hilbert spaces and let ''N'' : ''X'' → ''Y'' be a continuous linear map. Suppose that <math>N = UR</math> where ''R'' : ''X'' → ''X'' is the square-root of <math>N^* N</math> and ''U'' : ''X'' → ''Y'' is such that <math>U\big\vert_{\operatorname{Im} R} : \operatorname{Im} R \to \operatorname{Im} N</math> is a surjective isometry. 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 self-adjoint operators ''R''.{{sfn | Trèves | 2006 | pp=488-492}}

=== 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.{{sfn | Trèves | 2006 | pp=492-494}}
#''R'' : ''X'' → ''X'' is compact and <math>\operatorname{Tr} R</math> is finite, in which case <math>\operatorname{Tr} R = \| N \|_{\operatorname{Tr}}</math>.{{sfn | Trèves | 2006 | pp=492-494}}
#* Here, <math>\operatorname{Tr} R</math> 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 <math>\lambda_1 > \lambda_2 > \cdots</math> of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces <math>V_1, V_2, \ldots</math> such that the orthogonal (in ''H'') of <math>\operatorname{span}\left( V_1 \cup V_2 \cup \cdots \right)</math> is equal to <math>\ker R</math> (and hence also to <math>\ker N</math>) and for all ''k'', <math>R(x) = \lambda_k x</math> for all <math>x \in V_k</math>; the trace is defined as <math display="inline">\operatorname{Tr} R := \sum_{k} \lambda_k \dim V_k</math>.  
#<math>{}^{t}N : Y'_{b} \to X'_{b}</math> is nuclear, in which case <math>\| {}^{t}N \|_{\operatorname{Tr}} = \| N \|_{\operatorname{Tr}}</math>. {{sfn | Trèves | 2006 | p=484}}
#There are two orthogonal sequences <math>\left( x_i \right)_{i=1}^{\infty}</math> in ''X'' and <math>\left( y_i \right)_{i=1}^{\infty}</math> in ''Y'', and a sequence <math>\left( \lambda_i \right)_{i=1}^{\infty}</math> in <math>l^1</math> such that for all <math>x \in X</math>, <math display="inline">N(x) = \sum_{i} \lambda_i \langle x, x_i \rangle y_i</math>.{{sfn | Trèves | 2006 | pp=492-494}}
#''N'' : ''X'' → ''Y'' is an integral map.{{sfn | Trèves | 2006 | pp=502-508}}

== 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 <math display="inline">p_U(x) = \inf_{r > 0, x \in r U} r</math> and let <math>\pi : X \to X/p_U^{-1}(0)</math> be the canonical projection. One can define the auxiliary Banach space <math>\hat{X}_U</math> with the canonical map <math>\hat{\pi}_U : X \to \hat{X}_U</math> whose image, <math>X/p_U^{-1}(0)</math>, is dense in <math>\hat{X}_U</math> as well as the auxiliary space <math>F_B = \operatorname{span} B</math> normed by <math display="inline">p_B(y) = \inf_{r > 0, y \in r B} r</math> and with a canonical map <math>\iota : F_B \to F</math> being the (continuous) canonical injection. 
Given any continuous linear map <math>T : \hat{X}_U \to Y_B</math> one obtains through composition the continuous linear map <math>\hat{\pi}_U \circ T \circ \iota : X \to Y</math>; thus we have an injection <math display="inline">L \left( \hat{X}_U; Y_B \right) \to L(X; Y)</math> and we henceforth use this map to identify <math display="inline">L \left( \hat{X}_U; Y_B \right)</math> as a subspace of <math>L(X; Y)</math>.{{sfn | Trèves | 2006 | pp=478-479}}

'''Definition''': Let ''X'' and ''Y'' be Hausdorff locally convex spaces. The union of all <math display="inline">L^1\left( \hat{X}_U; Y_B \right)</math> 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 <math>L^1(X; Y)</math> and its elements are call '''nuclear mappings''' of ''X'' into ''Y''.{{sfn | Trèves | 2006 | pp=478-479}}

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, <math>N : X \to Y</math> a nuclear map, and <math>M : W \to X</math> and <math>P : Y \to Z</math> be continuous linear maps. Then <math>N \circ M : W \to Y</math>, <math>P \circ N : X \to Z</math>, and <math>P \circ N \circ M : W \to Z</math> are nuclear and if in addition ''W'', ''X'', ''Y'', and ''Z'' are all Banach spaces then <math display="inline">\left\| P \circ N \circ M\right\|_{\operatorname{Tr}} \leq \left\| P \right\| \left\| N \right\|_{\operatorname{Tr}} \| \left\| M \right\|</math>.{{sfn | Trèves | 2006 | pp=479-481}}{{sfn | Schaefer|Wolff| 1999 | p=100}} 
* If <math>N : X \to Y</math> is a nuclear map between two Hausdorff locally convex spaces, then its transpose <math>{}^{t}N : Y'_{b} \to X'_{b}</math> is a continuous nuclear map (when the dual spaces carry their strong dual topologies).{{sfn | Trèves | 2006 | p=483}} 
** If in addition ''X'' and ''Y'' are Banach spaces, then <math display="inline">\left\| {}^{t}N \right\|_{\operatorname{Tr}} \leq \left\| N \right\|_{\operatorname{Tr}}</math>.{{sfn | Trèves | 2006 | p=484}}
* If <math>N : X \to Y</math> is a nuclear map between two Hausdorff locally convex spaces and if <math>\hat{X}</math> is a completion of ''X'', then the unique continuous extension <math>\hat{N} : \hat{X} \to Y</math> of ''N'' is nuclear.{{sfn | Schaefer|Wolff| 1999 | p=100}}

=== Characterizations ===
Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let <math>N : X \to Y</math> be a continuous linear operator. 
* The following are equivalent:
*# <math>N : X \to Y</math> 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 <math>N(U) \subseteq B</math> and the induced map <math>\overline{N}_0 : \hat{X}_U \to Y_B</math> is nuclear, where <math>\overline{N}_0</math> is the unique continuous extension of <math>N_0 : X_U \to Y_B</math>, which is the unique map satisfying <math>N = \operatorname{In}_B \circ N_0 \circ \pi_U</math> where <math>\operatorname{In}_B : Y_B \to Y</math> is the natural inclusion and <math>\pi_U : X \to X / p_U^{-1}(0)</math> is the canonical projection.{{sfn | Schaefer|Wolff| 1999 | p=98}}
*# There exist Banach spaces <math>B_1</math> and <math>B_2</math> and continuous linear maps <math>f : X \to B_1</math>, <math>n : B_1 \to B_2</math>, and <math>g : B_2 \to Y</math> such that <math>n : B_1 \to B_2</math> is nuclear and <math>N = g \circ n \circ f</math>.{{sfn | Trèves | 2006 | pp=481-483}}  
*# There exists an equicontinuous sequence <math>\left( x_i' \right)_{i=1}^{\infty}</math> in <math>X'</math>, a bounded Banach disk <math>B \subseteq Y</math>, a sequence <math>\left( y_i \right)_{i=1}^{\infty}</math> in ''B'', and a complex sequence <math>\left( c_i \right)_{i=1}^{\infty}</math> such that <math display="inline">\sum_{i=1}^{\infty} |c_i| < \infty</math> and <math>N</math> is equal to the mapping:{{sfn | Trèves | 2006 | pp=481-483}} <math display="inline">N(x) = \sum_{i=1}^{\infty} c_i x'_i(x) y_i</math> for all <math>x \in X</math>.
* If ''X'' is barreled and ''Y'' is quasi-complete, then ''N'' is nuclear if and only if ''N'' has a representation of the form <math display="inline">N(x) = \sum_{i=1}^{\infty} c_i x'_i(x) y_i</math> with <math>\left( x_i' \right)_{i=1}^{\infty}</math> bounded in <math>X'</math>, <math>\left( y_i \right)_{i=1}^{\infty}</math> bounded in ''Y'' and <math display="inline">\sum_{i=1}^{\infty} |c_i| < \infty</math>.{{sfn | Trèves | 2006 | pp=481-483}}

=== Properties ===
The following is a type of ''Hahn-Banach theorem'' for extending nuclear maps:
* If <math>E : X \to Z</math> is a TVS-embedding and <math>N : X \to Y</math> is a nuclear map then there exists a nuclear map <math>\tilde{N} : Z \to Y</math> such that <math>\tilde{N} \circ E = N</math>. Furthermore, when ''X'' and ''Y'' are Banach spaces and ''E'' is an isometry then for any <math>\epsilon > 0</math>, <math>\tilde{N}</math> can be picked so that <math>\| \tilde{N} \|_{\operatorname{Tr}} \leq \| N \|_{\operatorname{Tr}} + \epsilon</math>.{{sfn | Trèves | 2006 | p=485}} 
* Suppose that <math>E : X \to Z</math> is a TVS-embedding whose image is closed in ''Z'' and let <math>\pi : Z \to Z / \operatorname{Im} E</math> be the canonical projection. Suppose all that every compact disk in <math>Z / \operatorname{Im} E</math> is the image under <math>\pi</math> 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 <math>\operatorname{Im} E</math> is weakly closed in ''Z''). Then for every nuclear map <math>N : Y \to Z / \operatorname{Im} E</math> there exists a nuclear map <math>\tilde{N} : Y \to Z</math> such that <math>\pi \circ \tilde{N} = N</math>. 
** Furthermore, when ''X'' and ''Z'' are Banach spaces and ''E'' is an isometry then for any <math>\epsilon > 0</math>, <math>\tilde{N}</math> can be picked so that <math display="inline">\left\| \tilde{N} \right\|_{\operatorname{Tr}} \leq \left\| N \right\|_{\operatorname{Tr}} + \epsilon</math>.{{sfn | Trèves | 2006 | p=485}}

Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let <math>N : X \to Y</math> be a continuous linear operator. 
* Any nuclear map is compact.{{sfn | Trèves | 2006 | p=483}}
* For every topology of uniform convergence on <math>L( X; Y )</math>, the nuclear maps are contained in the closure of <math>X' \otimes Y</math> (when <math>X' \otimes Y</math> is viewed as a subspace of <math>L( X; Y )</math>).{{sfn | Schaefer|Wolff| 1999 | p=98}}

== See also ==

* {{annotated link|Covariance operator}}
* {{annotated link|Initial topology}}
* {{annotated link|Inductive tensor product}}
* {{annotated link|Injective tensor product}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Nuclear operators between Banach spaces}}
* {{annotated link|Nuclear space}}
* {{annotated link|Projective tensor product}}
* {{annotated link|Tensor product of Hilbert spaces}}
* {{annotated link|Topological tensor product}}
* {{annotated link|Trace class}}
* {{annotated link|Topological vector space}}

== References ==
{{Reflist}}

==Bibliography==
* {{cite book | last=Diestel | first=Joe | title=The metric theory of tensor products : Grothendieck's résumé revisited | publisher=American Mathematical Society | location=Providence, R.I | year=2008 | isbn=978-0-8218-4440-3 | oclc=185095773 }} <!-- {{sfn | Diestel | 2008 | p=}} -->
* {{cite book | last=Dubinsky | first=Ed | title=The structure of nuclear Fréchet spaces | publisher=Springer-Verlag | location=Berlin New York | year=1979 | isbn=3-540-09504-7 | oclc=5126156 }} <!-- {{sfn | Dubinsky | 1979 | p=}} -->
* {{cite book | last=Grothendieck | first=Alexander | title=Produits tensoriels topologiques et espaces nucléaires | publisher=American Mathematical Society | location=Providence | year=1966 | isbn=0-8218-1216-5 | oclc=1315788 | language=fr }} <!-- {{sfn | Grothendieck | 1966 | p=}} -->
* {{cite book | last=Husain | first=Taqdir | title=Barrelledness in topological and ordered vector spaces | publisher=Springer-Verlag | location=Berlin New York | year=1978 | isbn=3-540-09096-7 | oclc=4493665 }} <!-- {{sfn | Husain | 1978 | p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} --> 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{cite book | last=Nlend | first=H | title=Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis | publisher=North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland | location=Amsterdam New York New York | year=1977 | isbn=0-7204-0712-5 | oclc=2798822 }} <!-- {{sfn | Nlend | 1977 | p=}} -->
* {{cite book | last=Nlend | first=H | title=Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality | publisher=North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland | location=Amsterdam New York New York, N.Y | year=1981 | isbn=0-444-86207-2 | oclc=7553061 }} <!-- {{sfn | Nlend | 1981 | p=}} -->
* {{cite book | last=Pietsch | first=Albrecht | title=Nuclear locally convex spaces | publisher=Springer-Verlag | location=Berlin,New York | year=1972 | isbn=0-387-05644-0 | oclc=539541 }} <!-- {{sfn | Pietsch | 1972 | p=}} -->
* {{cite book | last=Robertson | first=A. P. | title=Topological vector spaces | publisher=University Press | location=Cambridge England | year=1973 | isbn=0-521-29882-2 | oclc=589250 }} <!-- {{sfn | Robertson | 1973 | p=}} -->
* {{cite book | last=Ryan | first=Raymond | title=Introduction to tensor products of Banach spaces | publisher=Springer | location=London New York | year=2002 | isbn=1-85233-437-1 | oclc=48092184 }} <!-- {{sfn | Ryan | 2002 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* {{cite book | last=Wong | title=Schwartz spaces, nuclear spaces, and tensor products | publisher=Springer-Verlag | location=Berlin New York | year=1979 | isbn=3-540-09513-6 | oclc=5126158 }} <!-- {{sfn | Wong | 1979 | p=}} -->

== External links ==

* [https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

{{Functional analysis}}
{{TopologicalTensorProductsAndNuclearSpaces}}

<!--- Categories --->


[[Category:Topological vector spaces]]
[[Category:Tensors]]
[[Category:Operator theory]]
[[Category:Linear operators]]

{{Sourceattribution|Nuclear operator}}