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In mathematics,nuclear operators are an important class of linear operators introduced byAlexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to theprojective tensor product of twotopological vector spaces (TVSs).

Throughout letX,Y, andZ betopological vector spaces (TVSs) andL :X →Y be a linear operator (no assumption of continuity is made unless otherwise stated).

• Theprojective tensor product of twolocally convex TVSsX andY is denoted by${\displaystyle X{\otimes }_{\pi }Y}$ and the completion of this space will be denoted by${\displaystyle X{\hat{\otimes }}_{\pi }Y}$.
• L :X →Y is atopological homomorphism orhomomorphism, if it is linear, continuous, and${\displaystyle L:X\to \mathrm{Im}L}$ is anopen map, where${\displaystyle \mathrm{Im}L}$, the image ofL, has thesubspace topology induced byY.
  • IfS is a subspace ofX then both the quotient mapX →X/S and the canonical injectionS →X are homomorphisms.
• The set of continuous linear mapsX →Z (resp. continuous bilinear maps${\displaystyle X\times Y\to Z}$) will be denoted by L(X,Z) (resp. B(X,Y;Z)) where ifZ is the underlying scalar field then we may instead write L(X) (resp. B(X,Y)).
• Any linear map${\displaystyle L:X\to Y}$ can be canonically decomposed as follows:${\displaystyle X\to X/\ker L\stackrel{L_0}{\to }\mathrm{Im}L\to Y}$ where${\displaystyle L_0\left(x+\ker L\right):=L(x)}$ defines abijection called thecanonical bijection associated withL.
• X* or${\displaystyle X^{\prime }}$ will denote the continuous dual space ofX.
  • To increase the clarity of the exposition, we use the common convention of writing elements of${\displaystyle X^{\prime }}$ with a prime following the symbol (e.g.${\displaystyle x^{\prime }}$ denotes an element of${\displaystyle X^{\prime }}$ and not, say, a derivative and the variablesx and${\displaystyle x^{\prime }}$ need not be related in any way).
• ${\displaystyle X^{\mathrm{\#}}}$ will denote thealgebraic dual space ofX (which is the vector space of all linear functionals onX, whether continuous or not).
• A linear mapL :H →H from a Hilbert space into itself is calledpositive if${\displaystyle ⟨L(x),x⟩\ge 0}$ for every${\displaystyle x\in H}$. In this case, there is a unique positive mapr :H →H, called thesquare-root ofL, such that${\displaystyle L=r\circ r}$.^[1]
  • If${\displaystyle L:H_1\to H_2}$ is any continuous linear map between Hilbert spaces, then${\displaystyle L^{\ast }\circ L}$ is always positive. Now letR :H →H denote its positive square-root, which is called theabsolute value ofL. Define${\displaystyle U:H_1\to H_2}$ first on${\displaystyle \mathrm{Im}R}$ by setting${\displaystyle U(x)=L(x)}$ for${\displaystyle x=R\left(x_1\right)\in \mathrm{Im}R}$ and extending${\displaystyle U}$ continuously to${\displaystyle \overline{\mathrm{Im}R}}$, and then defineU on${\displaystyle \ker R}$ by setting${\displaystyle U(x)=0}$ for${\displaystyle x\in \ker R}$ and extend this map linearly to all of${\displaystyle H_1}$. The map${\displaystyle U{|}_{\mathrm{Im}R}:\mathrm{Im}R\to \mathrm{Im}L}$ is a surjective isometry and${\displaystyle L=U\circ R}$.
• A linear map${\displaystyle \mathrm{\Lambda }:X\to Y}$ is calledcompact orcompletely continuous if there is a neighborhoodU of the origin inX such that${\displaystyle \mathrm{\Lambda }(U)}$ isprecompact inY.^[2]

In a Hilbert space, positive compact linear operators, sayL :H →H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:^[3]

There is a sequence of positive numbers, decreasing and either finite or else converging to 0,${\displaystyle r_1>r_2>\cdots >r_k>\cdots }$ and a sequence of nonzero finite dimensional subspaces${\displaystyle V_i}$ ofH (i = 1, 2,${\displaystyle \dots }$) with the following properties: (1) the subspaces${\displaystyle V_i}$ are pairwise orthogonal; (2) for everyi and every${\displaystyle x\in V_i}$,${\displaystyle L(x)=r_{i}x}$; and (3) the orthogonal of the subspace spanned by$\bigcup _i{V}_i$ is equal to the kernel ofL.^[3]

• σ(X,X′) denotes thecoarsest topology onX making every map inX′ continuous and${\displaystyle X_{\sigma \left(X,X^{\prime }\right)}}$ or${\displaystyle X_{\sigma }}$ denotesX endowed with this topology.
• σ(X′,X) denotesweak-* topology on X* and${\displaystyle X_{\sigma \left(X^{\prime },X\right)}}$ or${\displaystyle X_{\sigma }^{\prime }}$ denotesX′ endowed with this topology.
  • Note that every${\displaystyle x_0\in X}$ induces a map${\displaystyle X^{\prime }\to \mathbb{R}}$ defined by${\displaystyle \lambda \mapsto \lambda (x_0)}$.σ(X′, X) is the coarsest topology on X′ making all such maps continuous.
• b(X, X′) denotes thetopology of bounded convergence onX and${\displaystyle X_{b\left(X,X^{\prime }\right)}}$ or${\displaystyle X_b}$ denotesX endowed with this topology.
• b(X′, X) denotes thetopology of bounded convergence on X′ or thestrong dual topology on X′ and${\displaystyle X_{b\left(X^{\prime },X\right)}}$ or${\displaystyle X_b^{\prime }}$ denotesX′ endowed with this topology.
  • As usual, ifX* 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).

LetX andY be vector spaces (no topology is needed yet) and let Bi(X,Y) be the space of allbilinear maps defined on${\displaystyle X\times Y}$ and going into the underlying scalar field.

For every${\displaystyle (x,y)\in X\times Y}$, let${\displaystyle {\chi }_{(x,y)}}$ be the canonicallinear form on Bi(X,Y) defined by${\displaystyle {\chi }_{(x,y)}(u):=u(x,y)}$ for everyu ∈ Bi(X,Y). This induces a canonical map${\displaystyle \chi :X\times Y\to \mathrm{B}\mathrm{i}(X,Y{)}^{\mathrm{\#}}}$ defined by${\displaystyle \chi (x,y):={\chi }_{(x,y)}}$, where${\displaystyle \mathrm{B}\mathrm{i}(X,Y{)}^{\mathrm{\#}}}$ denotes thealgebraic dual of Bi(X,Y). If we denote the span of the range of𝜒 byX ⊗Y then it can be shown thatX ⊗Y together with𝜒 forms atensor product ofX andY (wherex ⊗y :=𝜒(x,y)). This gives us a canonical tensor product ofX andY.

IfZ is any other vector space then the mapping Li(X ⊗Y;Z) → Bi(X,Y;Z) given byu ↦u ∘𝜒 is an isomorphism of vector spaces. In particular, this allows us to identify thealgebraic dual ofX ⊗Y with the space of bilinear forms onX ×Y.^[4] Moreover, ifX andY are locally convextopological vector spaces (TVSs) and ifX ⊗Y is given theπ-topology then for every locally convex TVSZ, this map restricts to a vector space isomorphism${\displaystyle L(X{\otimes }_{\pi }Y;Z)\to B(X,Y;Z)}$ from the space ofcontinuous linear mappings onto the space ofcontinuous bilinear mappings.^[5] In particular, the continuous dual ofX ⊗Y can be canonically identified with the space B(X,Y) of continuous bilinear forms onX ×Y; furthermore, under this identification theequicontinuous subsets of B(X,Y) are the same as the equicontinuous subsets of${\displaystyle (X{\otimes }_{\pi }Y{)}^{\prime }}$.^[5]

There is a canonical vector space embedding${\displaystyle I:X^{\prime }\otimes Y\to L(X;Y)}$ defined by sending$z:=\sum _i^n{x}_i^{\prime }\otimes y_i$ to the map

${\displaystyle x\mapsto \sum _i^n{x}_i^{\prime }(x)y_i.}$

Assuming thatX andY are Banach spaces, then the map${\displaystyle I:X_b^{\prime }{\otimes }_{\pi }Y\to L_b(X;Y)}$ has norm${\displaystyle 1}$ (to see that the norm is${\displaystyle \le 1}$, note that$\Vert I(z)\Vert =\underset{\Vert x\Vert \le 1}{sup}\Vert I(z)(x)\Vert =\underset{\Vert x\Vert \le 1}{sup}\Vert \sum _{i=1}^n{x}_i^{\prime }(x)y_i\Vert \le \underset{\Vert x\Vert \le 1}{sup}\sum _{i=1}^n\Vert x_i^{\prime }\Vert \Vert x\Vert \Vert y_i\Vert \le \sum _{i=1}^n\Vert x_i^{\prime }\Vert \Vert y_i\Vert$ so that${\displaystyle \Vert I(z)\Vert \le {\Vert z\Vert }_{\pi }}$). Thus it has a continuous extension to a map${\displaystyle \hat{I}:X_b^{\prime }{\hat{\otimes }}_{\pi }Y\to L_b(X;Y)}$, where it is known that this map is not necessarily injective.^[6] The range of this map is denoted by${\displaystyle L^1(X;Y)}$ and its elements are callednuclear operators.^[7]${\displaystyle L^1(X;Y)}$ is TVS-isomorphic to${\displaystyle \left(X_b^{\prime }{\hat{\otimes }}_{\pi }Y\right)/\ker \hat{I}}$ and the norm on this quotient space, when transferred to elements of${\displaystyle L^1(X;Y)}$ via the induced map${\displaystyle \hat{I}:\left(X_b^{\prime }{\hat{\otimes }}_{\pi }Y\right)/\ker \hat{I}\to L^1(X;Y)}$, is called thetrace-norm and is denoted by${\displaystyle \Vert \cdot {\Vert }_{\mathrm{Tr}}}$. Explicitly,^{[clarification needed explicitly or especially?]} if${\displaystyle T:X\to Y}$ is a nuclear operator then${\Vert T\Vert }_{\mathrm{Tr}}:=\underset{z\in {\hat{I}}^{-1}\left(T\right)}{inf}{\Vert z\Vert }_{\pi }$.

Suppose thatX andY are Banach spaces and that${\displaystyle N:X\to Y}$ is a continuous linear operator.

• The following are equivalent:
  1. ${\displaystyle N:X\to Y}$ is nuclear.
  2. There exists a sequence${\displaystyle {\left(x_i^{\prime }\right)}_{i=1}^{\mathrm{\infty }}}$ in the closed unit ball of${\displaystyle X^{\prime }}$, a sequence${\displaystyle {\left(y_i\right)}_{i=1}^{\mathrm{\infty }}}$ in the closed unit ball of${\displaystyle Y}$, and a complex sequence${\displaystyle {\left(c_i\right)}_{i=1}^{\mathrm{\infty }}}$ such that and${\displaystyle N}$ is equal to the mapping:^[8]$N(x)=\sum _{i=1}^{\mathrm{\infty }}{c}_i{x}_i^{\prime }(x)y_i$ for all${\displaystyle x\in X}$. Furthermore, the trace-norm${\displaystyle \Vert N{\Vert }_{\mathrm{Tr}}}$ is equal to the infimum of the numbers$\sum _{i=1}^{\mathrm{\infty }}|c_i|$ over the set of all representations of${\displaystyle N}$ as such a series.^[8]
• IfY isreflexive then${\displaystyle N:X\to Y}$ is a nuclear if and only if${\displaystyle {}^{t}N:Y_b^{\prime }\to X_b^{\prime }}$ is nuclear, in which case${\Vert {}^{t}N\Vert }_{\mathrm{Tr}}={\Vert N\Vert }_{\mathrm{Tr}}$.^[9]

LetX andY be Banach spaces and let${\displaystyle N:X\to Y}$ be a continuous linear operator.

• If${\displaystyle N:X\to Y}$ is a nuclear map then its transpose${\displaystyle {}^{t}N:Y_b^{\prime }\to X_b^{\prime }}$ is a continuous nuclear map (when the dual spaces carry their strong dual topologies) and${\Vert {}^{t}N\Vert }_{\mathrm{Tr}}\le {\Vert N\Vert }_{\mathrm{Tr}}$.^[10]

Nuclearautomorphisms of aHilbert space are calledtrace class operators.

LetX andY be Hilbert spaces and letN :X →Y be a continuous linear map. Suppose that${\displaystyle N=UR}$ whereR :X →X is the square-root of${\displaystyle N^{\ast }N}$ andU :X →Y is such that${\displaystyle U{|}_{\mathrm{Im}R}:\mathrm{Im}R\to \mathrm{Im}N}$ is a surjective isometry. ThenN is a nuclear map if and only ifR is a nuclear map; hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive self-adjoint operatorsR.^[11]

LetX andY be Hilbert spaces and letN :X →Y be a continuous linear map whose absolute value isR :X →X. The following are equivalent:

1. N :X →Y is nuclear.
2. R :X →X is nuclear.^[12]
3. R :X →X is compact and${\displaystyle \mathrm{Tr}R}$ is finite, in which case${\displaystyle \mathrm{Tr}R=\Vert N{\Vert }_{\mathrm{Tr}}}$.^[12]
   • Here,${\displaystyle \mathrm{Tr}R}$ is thetrace ofR and it is defined as follows: SinceR is a continuous compact positive operator, there exists a (possibly finite) sequence${\displaystyle {\lambda }_1>{\lambda }_2>\cdots }$ of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces${\displaystyle V_1,V_2,\dots }$ such that the orthogonal (inH) of${\displaystyle \mathrm{span}\left(V_1\cup V_2\cup \cdots \right)}$ is equal to${\displaystyle \ker R}$ (and hence also to${\displaystyle \ker N}$) and for allk,${\displaystyle R(x)={\lambda }_{k}x}$ for all${\displaystyle x\in V_k}$; the trace is defined as$\mathrm{Tr}R:=\sum _k{\lambda }_k\dim V_k$.
4. ${\displaystyle {}^{t}N:Y_b^{\prime }\to X_b^{\prime }}$ is nuclear, in which case${\displaystyle \Vert {}^{t}N{\Vert }_{\mathrm{Tr}}=\Vert N{\Vert }_{\mathrm{Tr}}}$.^[9]
5. There are two orthogonal sequences${\displaystyle (x_i{)}_{i=1}^{\mathrm{\infty }}}$ inX and${\displaystyle (y_i{)}_{i=1}^{\mathrm{\infty }}}$ inY, and a sequence${\displaystyle {\left({\lambda }_i\right)}_{i=1}^{\mathrm{\infty }}}$ in${\displaystyle {\ell }^1}$ such that for all${\displaystyle x\in X}$,$N(x)=\sum _i{\lambda }_i⟨x,x_i⟩y_i$.^[12]
6. N :X →Y is anintegral map.^[13]

Suppose thatU is a convex balanced closed neighborhood of the origin inX andB is a convex balanced boundedBanach disk inY with bothX andY locally convex spaces. Let$p_U(x)=\underset{r>0,x\in rU}{inf}r$ and let${\displaystyle \pi :X\to X/p_U^{-1}(0)}$ be the canonical projection. One can define theauxiliary Banach space${\displaystyle {\hat{X}}_U}$ with the canonical map${\displaystyle {\hat{\pi }}_U:X\to {\hat{X}}_U}$ whose image,${\displaystyle X/p_U^{-1}(0)}$, is dense in${\displaystyle {\hat{X}}_U}$ as well as the auxiliary space${\displaystyle F_B=\mathrm{span}B}$ normed by$p_B(y)=\underset{r>0,y\in rB}{inf}r$ and with a canonical map${\displaystyle \iota :F_B\to F}$ being the (continuous) canonical injection. Given any continuous linear map${\displaystyle T:{\hat{X}}_U\to Y_B}$ one obtains through composition the continuous linear map${\displaystyle {\hat{\pi }}_U\circ T\circ \iota :X\to Y}$; thus we have an injection$L\left({\hat{X}}_U;Y_B\right)\to L(X;Y)$ and we henceforth use this map to identify$L\left({\hat{X}}_U;Y_B\right)$ as a subspace of${\displaystyle L(X;Y)}$.^[7]

Definition: LetX andY be Hausdorff locally convex spaces. The union of all$L^1\left({\hat{X}}_U;Y_B\right)$ asU ranges over all closed convex balanced neighborhoods of the origin inX andB ranges over all boundedBanach disks inY, is denoted by${\displaystyle L^1(X;Y)}$ and its elements are callnuclear mappings ofX intoY.^[7]

WhenX andY are Banach spaces, then this new definition ofnuclear mapping is consistent with the original one given for the special case whereX andY are Banach spaces.

• LetW,X,Y, andZ be Hausdorff locally convex spaces,${\displaystyle N:X\to Y}$ a nuclear map, and${\displaystyle M:W\to X}$ and${\displaystyle P:Y\to Z}$ be continuous linear maps. Then${\displaystyle N\circ M:W\to Y}$,${\displaystyle P\circ N:X\to Z}$, and${\displaystyle P\circ N\circ M:W\to Z}$ are nuclear and if in additionW,X,Y, andZ are all Banach spaces then${\Vert P\circ N\circ M\Vert }_{\mathrm{Tr}}\le \Vert P\Vert {\Vert N\Vert }_{\mathrm{Tr}}\Vert \Vert M\Vert$.^[14][15]
• If${\displaystyle N:X\to Y}$ is a nuclear map between two Hausdorff locally convex spaces, then its transpose${\displaystyle {}^{t}N:Y_b^{\prime }\to X_b^{\prime }}$ is a continuous nuclear map (when the dual spaces carry their strong dual topologies).^[2]
  • If in additionX andY are Banach spaces, then${\Vert {}^{t}N\Vert }_{\mathrm{Tr}}\le {\Vert N\Vert }_{\mathrm{Tr}}$.^[9]
• If${\displaystyle N:X\to Y}$ is a nuclear map between two Hausdorff locally convex spaces and if${\displaystyle \hat{X}}$ is a completion ofX, then the unique continuous extension${\displaystyle \hat{N}:\hat{X}\to Y}$ ofN is nuclear.^[15]

LetX andY be Hausdorff locally convex spaces and let${\displaystyle N:X\to Y}$ be a continuous linear operator.

• The following are equivalent:
  1. ${\displaystyle N:X\to Y}$ is nuclear.
  2. (Definition) There exists a convex balanced neighborhoodU of the origin inX and a boundedBanach diskB inY such that${\displaystyle N(U)\subseteq B}$ and the induced map${\displaystyle {\overline{N}}_0:{\hat{X}}_U\to Y_B}$ is nuclear, where${\displaystyle {\overline{N}}_0}$ is the unique continuous extension of${\displaystyle N_0:X_U\to Y_B}$, which is the unique map satisfying${\displaystyle N={\mathrm{In}}_B\circ N_0\circ {\pi }_U}$ where${\displaystyle {\mathrm{In}}_B:Y_B\to Y}$ is the natural inclusion and${\displaystyle {\pi }_U:X\to X/p_U^{-1}(0)}$ is the canonical projection.^[6]
  3. There exist Banach spaces${\displaystyle B_1}$ and${\displaystyle B_2}$ and continuous linear maps${\displaystyle f:X\to B_1}$,${\displaystyle n:B_1\to B_2}$, and${\displaystyle g:B_2\to Y}$ such that${\displaystyle n:B_1\to B_2}$ is nuclear and${\displaystyle N=g\circ n\circ f}$.^[8]
  4. There exists an equicontinuous sequence${\displaystyle {\left(x_i^{\prime }\right)}_{i=1}^{\mathrm{\infty }}}$ in${\displaystyle X^{\prime }}$, a boundedBanach disk${\displaystyle B\subseteq Y}$, a sequence${\displaystyle {\left(y_i\right)}_{i=1}^{\mathrm{\infty }}}$ inB, and a complex sequence${\displaystyle {\left(c_i\right)}_{i=1}^{\mathrm{\infty }}}$ such that and${\displaystyle N}$ is equal to the mapping:^[8]$N(x)=\sum _{i=1}^{\mathrm{\infty }}{c}_i{x}_i^{\prime }(x)y_i$ for all${\displaystyle x\in X}$.
• IfX is barreled andY isquasi-complete, thenN is nuclear if and only ifN has a representation of the form$N(x)=\sum _{i=1}^{\mathrm{\infty }}{c}_i{x}_i^{\prime }(x)y_i$ with${\displaystyle {\left(x_i^{\prime }\right)}_{i=1}^{\mathrm{\infty }}}$ bounded in${\displaystyle X^{\prime }}$,${\displaystyle {\left(y_i\right)}_{i=1}^{\mathrm{\infty }}}$ bounded inY and.^[8]

The following is a type ofHahn-Banach theorem for extending nuclear maps:

• If${\displaystyle E:X\to Z}$ is a TVS-embedding and${\displaystyle N:X\to Y}$ is a nuclear map then there exists a nuclear map${\displaystyle \tilde{N}:Z\to Y}$ such that${\displaystyle \tilde{N}\circ E=N}$. Furthermore, whenX andY are Banach spaces andE is an isometry then for any${\displaystyle ϵ>0}$,${\displaystyle \tilde{N}}$ can be picked so that${\displaystyle \Vert \tilde{N}{\Vert }_{\mathrm{Tr}}\le \Vert N{\Vert }_{\mathrm{Tr}}+ϵ}$.^[16]
• Suppose that${\displaystyle E:X\to Z}$ is a TVS-embedding whose image is closed inZ and let${\displaystyle \pi :Z\to Z/\mathrm{Im}E}$ be the canonical projection. Suppose all that every compact disk in${\displaystyle Z/\mathrm{Im}E}$ is the image under${\displaystyle \pi }$ of a bounded Banach disk inZ (this is true, for instance, ifX andZ are both Fréchet spaces, or ifZ is the strong dual of a Fréchet space and${\displaystyle \mathrm{Im}E}$ is weakly closed inZ). Then for every nuclear map${\displaystyle N:Y\to Z/\mathrm{Im}E}$ there exists a nuclear map${\displaystyle \tilde{N}:Y\to Z}$ such that${\displaystyle \pi \circ \tilde{N}=N}$.
  • Furthermore, whenX andZ are Banach spaces andE is an isometry then for any${\displaystyle ϵ>0}$,${\displaystyle \tilde{N}}$ can be picked so that${\Vert \tilde{N}\Vert }_{\mathrm{Tr}}\le {\Vert N\Vert }_{\mathrm{Tr}}+ϵ$.^[16]

LetX andY be Hausdorff locally convex spaces and let${\displaystyle N:X\to Y}$ be a continuous linear operator.

• Any nuclear map is compact.^[2]
• For every topology of uniform convergence on${\displaystyle L(X;Y)}$, the nuclear maps are contained in the closure of${\displaystyle X^{\prime }\otimes Y}$ (when${\displaystyle X^{\prime }\otimes Y}$ is viewed as a subspace of${\displaystyle L(X;Y)}$).^[6]

• Auxiliary normed spaces
• Covariance operator – Operator in probability theory
• Initial topology – Coarsest topology making certain functions continuous
• Inductive tensor product
• Injective tensor product
• Locally convex topological vector space – Space with topology generated by convex sets
• Nuclear operators between Banach spaces
• Nuclear space – Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces
• Trace class – Compact operator for which a finite trace can be defined
• Topological vector space – Vector space with a notion of nearness

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• Grothendieck, Alexander (1966).Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society.ISBN 0-8218-1216-5.OCLC 1315788.
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• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
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• Nlend, H (1981).Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland.ISBN 0-444-86207-2.OCLC 7553061.
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• Nuclear space at ncatlab

• Topological vector spaces
• Tensors
• Operator theory
• Topological tensor products
• Linear operators

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