Projective tensor product

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The strongest locally convex topological vector space (TVS) topology on ${\displaystyle X\otimes Y,}$ the tensor product of two locally convex TVSs, making the canonical map ${\displaystyle \cdot \otimes \cdot :X\times Y\to X\otimes Y}$ (defined by sending ${\displaystyle (x,y)\in X\times Y}$ to ${\displaystyle x\otimes y}$) continuous is called the projective topology or the π-topology. When ${\displaystyle X\otimes Y}$ is endowed with this topology then it is denoted by ${\displaystyle X\otimes _{\pi }Y}$ and called the projective tensor product of ${\displaystyle X}$ and ${\displaystyle Y.}$

Preliminaries[]

Throughout let ${\displaystyle X,Y,}$ and ${\displaystyle Z}$ be topological vector spaces and ${\displaystyle L:X\to Y}$ be a linear map.

• ${\displaystyle L:X\to Y}$ is a topological homomorphism or homomorphism, if it is linear, continuous, and ${\displaystyle L:X\to \operatorname {Im} L}$ is an open map, where ${\displaystyle \operatorname {Im} L,}$ the image of ${\displaystyle L,}$ has the subspace topology induced by ${\displaystyle Y.}$
  • If ${\displaystyle S\subseteq X}$ is a subspace of ${\displaystyle X}$ then both the quotient map ${\displaystyle X\to X/S}$ and the canonical injection ${\displaystyle S\to X}$ are homomorphisms. In particular, any linear map ${\displaystyle L:X\to Y}$ can be canonically decomposed as follows: ${\displaystyle X\to X/\operatorname {ker} L{\overset {L_{0}}{\rightarrow }}\operatorname {Im} L\to Y}$ where ${\displaystyle L_{0}(x+\ker L):=L(x)}$ defines a bijection.
• The set of continuous linear maps ${\displaystyle X\to Z}$ (resp. continuous bilinear maps ${\displaystyle X\times Y\to Z}$) will be denoted by ${\displaystyle L(X;Z)}$ (resp. ${\displaystyle B(X,Y;Z)}$) where if ${\displaystyle Z}$ is the scalar field then we may instead write ${\displaystyle L(X)}$ (resp. ${\displaystyle B(X,Y)}$).
• We will denote the continuous dual space of ${\displaystyle X}$ by ${\displaystyle X^{\prime }}$ and the algebraic dual space (which is the vector space of all linear functionals on ${\displaystyle X,}$ whether continuous or not) by ${\displaystyle X^{\#}.}$
  • 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 variables ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$ need not be related in any way).
• A linear map ${\displaystyle L:H\to H}$ from a Hilbert space into itself is called positive if ${\displaystyle \langle L(x),X\rangle \geq 0}$ for every ${\displaystyle x\in H.}$ In this case, there is a unique positive map ${\displaystyle r:H\to H,}$ called the square-root of ${\displaystyle L,}$ 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 ${\displaystyle L^{*}\circ L}$ is always positive. Now let ${\displaystyle R:H\to H}$ denote its positive square-root, which is called the absolute value of ${\displaystyle L.}$ Define ${\displaystyle U:H_{1}\to H_{2}}$ first on ${\displaystyle \operatorname {Im} R}$ by setting ${\displaystyle U(x)=L(x)}$ for ${\displaystyle x=R\left(x_{1}\right)\in \operatorname {Im} R}$ and extending ${\displaystyle U}$ continuously to ${\displaystyle {\overline {\operatorname {Im} R}},}$ and then define ${\displaystyle U}$ on ${\displaystyle \operatorname {ker} R}$ by setting ${\displaystyle U(x)=0}$ for ${\displaystyle x\in \operatorname {ker} R}$ and extend this map linearly to all of ${\displaystyle H_{1}.}$ The map ${\displaystyle U{\big \vert }_{\operatorname {Im} R}:\operatorname {Im} R\to \operatorname {Im} L}$ is a surjective isometry and ${\displaystyle L=U\circ R.}$
• A linear map ${\displaystyle \Lambda :X\to Y}$ is called compact or completely continuous if there is a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle \Lambda (U)}$ is precompact in ${\displaystyle Y.}$^[2]
  • In a Hilbert space, positive compact linear operators, say ${\displaystyle L:H\to H}$ 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 ${\displaystyle V_{i}}$ of ${\displaystyle H}$ (${\displaystyle i=1,2,\ldots }$) with the following properties: (1) the subspaces ${\displaystyle V_{i}}$ are pairwise orthogonal; (2) for every ${\displaystyle i}$ and every ${\displaystyle x\in V_{i},}$ ${\displaystyle L(x)=r_{i}x}$; and (3) the orthogonal of the subspace spanned by ${\displaystyle \cup _{i}V_{i}}$ is equal to the kernel of ${\displaystyle L.}$^[3]

Notation for topologies[]

• ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ denotes the coarsest topology on ${\displaystyle X}$ making every map in ${\displaystyle X^{\prime }}$ continuous and ${\displaystyle X_{\sigma \left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{\sigma }}$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle \sigma \left(X^{\prime },X\right)}$ denotes weak-* topology on ${\displaystyle X^{\prime }}$ and ${\displaystyle X_{\sigma \left(X^{\prime },X\right)}}$ or ${\displaystyle X_{\sigma }^{\prime }}$ denotes ${\displaystyle X^{\prime }}$ endowed with this topology.
  • Every ${\displaystyle x_{0}\in X}$ induces a map ${\displaystyle X^{\prime }\to \mathbb {R} }$ defined by ${\displaystyle \lambda \mapsto \lambda \left(x_{0}\right).}$ ${\displaystyle \sigma \left(X^{\prime },X\right)}$ is the coarsest topology on ${\displaystyle X^{\prime }}$ making all such maps continuous.
• ${\displaystyle b\left(X,X^{\prime }\right)}$ denotes the topology of bounded convergence on ${\displaystyle X}$ and ${\displaystyle X_{b\left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{b}}$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle b\left(X^{\prime },X\right)}$ denotes the topology of bounded convergence on ${\displaystyle X^{\prime }}$ or the strong dual topology on ${\displaystyle X^{\prime }}$ and ${\displaystyle X_{b\left(X^{\prime },X\right)}}$ or ${\displaystyle X_{b}^{\prime }}$ denotes ${\displaystyle X^{\prime }}$ endowed with this topology.
  • As usual, if ${\displaystyle X^{\prime }}$ 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 ${\displaystyle b\left(X^{\prime },X\right).}$

A canonical tensor product as a subspace of the dual of Bi(X, Y)[]

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

For every ${\displaystyle (x,y)\in X\times Y}$ define a canonical bilinear form by ${\displaystyle \chi _{(x,y)}}$ with domain ${\displaystyle \operatorname {Bi} (X,Y)}$ by ${\displaystyle \chi _{(x,y)}(u):=u(x,y)}$ for every ${\displaystyle u\in Bi(X,Y).}$ This induces a canonical map ${\displaystyle \chi :X\times Y\to \operatorname {Bi} (X,Y)^{\#}}$ defined by ${\displaystyle \chi (x,y)=\chi _{(x,y)},}$ where ${\displaystyle \operatorname {Bi} (X,Y)^{\#}}$ denotes the algebraic dual of ${\displaystyle \operatorname {Bi} (X,Y).}$ If we denote the span of the range of ${\displaystyle \chi }$ by ${\displaystyle X\otimes Y}$ then ${\displaystyle X\otimes Y}$ together with ${\displaystyle \chi }$ forms a tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$ (where ${\displaystyle x\otimes y=\chi (x,y)}$). This gives us a canonical tensor product of ${\displaystyle X}$ and ${\displaystyle Y.}$

If ${\displaystyle Z}$ is any other vector space then the mapping ${\displaystyle Li(X\otimes Y;Z)\to \operatorname {Bi} (X,Y;Z)}$ given by ${\displaystyle u\mapsto u\circ \chi }$ is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of ${\displaystyle X\otimes Y}$ with the space of bilinear forms on ${\displaystyle X\times Y.}$^[4] Moreover, if ${\displaystyle X}$ and ${\displaystyle Y}$ are locally convex topological vector spaces (TVSs) and if ${\displaystyle X\otimes Y}$ is given the ${\displaystyle \pi }$-topology then for every locally convex TVS ${\displaystyle Z,}$ this map restricts to a vector space isomorphism ${\displaystyle L(X\otimes _{\pi }Y;Z)\to B(X,Y;Z)}$ from the space of continuous linear mappings onto the space of continuous bilinear mappings.^[5] In particular, the continuous dual of ${\displaystyle X\otimes Y}$ can be canonically identified with the space ${\displaystyle \operatorname {B} (X,Y)}$ of continuous bilinear forms on ${\displaystyle X\times Y}$; furthermore, under this identification the equicontinuous subsets of ${\displaystyle \operatorname {B} (X,Y)}$ are the same as the equicontinuous subsets of ${\displaystyle (X\otimes _{\pi }Y)^{\prime }.}$^[5]

The projective tensor product[]

Tensor product of seminorms[]

Throughout we will let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally convex topological vector spaces (local convexity allows us to define useful topologies^[5]). If ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$ then ${\displaystyle C_{p}:=\{x\in X:p(x)\leq 1\}}$ will be its closed unit ball.

If ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$ and ${\displaystyle q}$ is a seminorm on ${\displaystyle Y}$ then we can define the tensor product of ${\displaystyle p}$ and ${\displaystyle q}$ to be the map ${\displaystyle p\otimes q}$ defined on ${\displaystyle X\otimes Y}$ by

where ${\displaystyle W}$ is the balanced convex hull of ${\displaystyle C_{p}\otimes C_{q}=\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}.}$ Given ${\displaystyle b}$ in ${\displaystyle X\otimes Y,}$ this can also be expressed as^[6] where the infimum is taken over all finite sequences ${\displaystyle x_{1},\ldots ,x_{n}\in X}$ and ${\displaystyle y_{1},\ldots ,y_{n}\in Y}$ (of the same length) such that ${\displaystyle b=x_{1}\otimes y_{1}+\cdots +x_{n}\otimes y_{n}}$ (recall that it may not be possible to express ${\displaystyle b}$ as a simple tensor). If ${\displaystyle b=x\otimes y}$ then we have The seminorm ${\displaystyle p\otimes q}$ is a norm if and only if both ${\displaystyle p}$ and ${\displaystyle q}$ are norms.^[7]

If the topology of ${\displaystyle X}$ (resp. ${\displaystyle Y}$) is given by the family of seminorms ${\displaystyle \left(p_{\alpha }\right)}$ (resp. ${\displaystyle \left(q_{\beta }\right)}$) then ${\displaystyle X\otimes _{\pi }Y}$ is a locally convex space whose topology is given by the family of all possible tensor products of the two families (i.e. by ${\displaystyle \left(p_{\alpha }\otimes q_{\beta }\right)}$). In particular, if ${\displaystyle X}$ and ${\displaystyle Y}$ are seminormed spaces with seminorms ${\displaystyle p}$ and ${\displaystyle q,}$ respectively, then ${\displaystyle X\otimes _{\pi }Y}$ is a seminormable space whose topology is defined by the seminorm ${\displaystyle p\otimes q.}$^[8] If ${\displaystyle (X,p)}$ and ${\displaystyle (Y,q)}$ are normed spaces then ${\displaystyle \left(X\otimes Y,p\otimes q\right)}$ is also a normed space, called the projective tensor product of ${\displaystyle (X,p)}$ and ${\displaystyle (Y,q),}$ where the topology induced by ${\displaystyle p\otimes q}$ is the same as the π-topology.^[8]

If ${\displaystyle W}$ is a convex subset of ${\displaystyle X\otimes Y}$ then ${\displaystyle W}$ is a neighborhood of 0 in ${\displaystyle X\otimes _{\pi }Y}$ if and only if the preimage of ${\displaystyle W}$ under the map ${\displaystyle (x,y)\mapsto x\otimes y}$ is a neighborhood of 0; equivalent, if and only if there exist open subsets ${\displaystyle U\subseteq X}$ and ${\displaystyle V\subseteq Y}$ such that this preimage contains ${\displaystyle U\otimes V:=\{u\otimes v:u\in U,v\in V\}.}$^[9] It follows that if ${\displaystyle \left(U_{\alpha }\right)}$ and ${\displaystyle \left(V_{\beta }\right)}$ are neighborhood bases of the origin in ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively, then the set of convex hulls of all possible set ${\displaystyle U_{\alpha }\otimes V_{\beta }}$ form a neighborhood basis of the origin in ${\displaystyle X\otimes _{\pi }Y.}$

Universal property[]

If ${\displaystyle \tau }$ is a locally convex TVS topology on ${\displaystyle X\otimes Y}$ (${\displaystyle X\otimes Y}$ with this topology will be denoted by ${\displaystyle X\otimes _{\tau }Y}$), then ${\displaystyle \tau }$ is equal to the π-topology if and only if it has the following property:^[10]

For every locally convex TVS ${\displaystyle Z,}$ if ${\displaystyle I}$ is the canonical map from the space of all bilinear mappings of the form ${\displaystyle X\times Y\to Z,}$ going into the space of all linear mappings of ${\displaystyle X\otimes Y\to Z,}$ then when the domain of ${\displaystyle I}$ is restricted to ${\displaystyle B\left(X,Y;Z\right)}$ then the range of this restriction is the space ${\displaystyle L\left(X\otimes _{\tau }Y;Z\right)}$ of continuous linear operators ${\displaystyle X\otimes _{\tau }Y\to Z.}$

In particular, the continuous dual space of ${\displaystyle X\otimes _{\pi }Y}$ is canonically isomorphic to the space ${\displaystyle B(X,Y),}$ the space of continuous bilinear forms on ${\displaystyle X\times Y.}$

The π-topology[]

Note that the canonical vector space isomorphism ${\displaystyle I:B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right)\to L\left(X\otimes _{\pi }Y;Z\right)}$ preserves equicontinuous subsets. Since ${\displaystyle B(X,Y)}$ is canonically isomorphic to the continuous dual of ${\displaystyle X\otimes _{\pi }Y,}$ place on ${\displaystyle X\otimes Y}$ the topology of uniform convergence on equicontinuous subsets of ${\displaystyle \left(X\otimes _{\pi }Y\right)^{\prime }}$; this topology is identical to the π-topology.^[10]

Preserved properties[]

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally convex TVSs.

• If both ${\displaystyle X}$ and ${\displaystyle Y}$ are Hausdorff (resp. locally convex, metrizable, semi-metrizable, normable, semi-normable) then so is ${\displaystyle X\otimes _{\pi }Y.}$

Completion[]

In general, the space ${\displaystyle X\otimes _{\pi }Y}$ is not complete, even if both ${\displaystyle X}$ and ${\displaystyle Y}$ are complete (in fact, if ${\displaystyle X}$ and ${\displaystyle Y}$ are both infinite-dimensional Banach spaces then ${\displaystyle X\otimes _{\pi }Y}$ is necessarily not complete^[11]). However, ${\displaystyle X\otimes _{\pi }Y}$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by ${\displaystyle X{\widehat {\otimes }}_{\pi }Y,}$ via a linear topological embedding. Explicitly, this means that there is a continuous linear injection ${\displaystyle \operatorname {In} :X\otimes _{\pi }Y\to X{\widehat {\otimes }}_{\pi }Y}$ whose image is dense in ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ and that is a TVS-isomorphism onto its image. Using this map, ${\displaystyle X\otimes _{\pi }Y}$ is identified as a subspace of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y.}$

The continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ is the same as that of ${\displaystyle X\otimes _{\pi }Y,}$ namely the space of continuous bilinear forms ${\displaystyle B(X,Y).}$:^[12]

Any continuous map on ${\displaystyle X\otimes _{\pi }Y}$ can be extended to a unique continuous map on ${\displaystyle X{\widehat {\otimes }}_{\pi }Y.}$ In particular, if ${\displaystyle u:X_{1}\to Y_{1}}$ and ${\displaystyle v:X_{2}\to Y_{2}}$ are continuous linear maps between locally convex spaces then their tensor product ${\displaystyle u\otimes v:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}\subseteq Y_{1}{\widehat {\otimes }}_{\pi }Y_{2},}$ which is necessarily continuous, can be extended to a unique continuous linear function ${\displaystyle u{\widehat {\otimes }}_{\pi }v:X_{1}{\widehat {\otimes }}_{\pi }X_{2}\to Y_{1}{\widehat {\otimes }}_{\pi }Y_{2},}$ which may also be denoted by ${\displaystyle u{\widehat {\otimes }}v}$ if no ambiguity would arise.

Note that if ${\displaystyle X}$ and ${\displaystyle Y}$ are metrizable then so are ${\displaystyle X\otimes _{\pi }Y}$ and ${\displaystyle X{\widehat {\otimes }}_{\pi }Y,}$ where in particular ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ will be an F-space.

Grothendieck's representation of elements of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$[]

In a Hausdorff locally convex space ${\displaystyle X,}$ a sequence ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ is absolutely convergent if ${\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty }$ for every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X.}$^[13] We write ${\displaystyle x=\sum _{i=1}^{\infty }x_{i}}$ if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }}$ converges to ${\displaystyle x}$ in ${\displaystyle X.}$^[13]

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.^[14]

The next theorem shows that it is possible to make the representation of ${\displaystyle z}$ independent of the sequences ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ and ${\displaystyle \left(y_{i}\right)_{i=1}^{\infty }.}$

Topology of bi-bounded convergence[]

Let ${\displaystyle {\mathfrak {B}}_{X}}$ and ${\displaystyle {\mathfrak {B}}_{Y}}$ denote the families of all bounded subsets of ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively. Since the continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ is the space of continuous bilinear forms ${\displaystyle B(X,Y),}$ we can place on ${\displaystyle B(X,Y)}$ the topology of uniform convergence on sets in ${\displaystyle {\mathfrak {B}}_{X}\times {\mathfrak {B}}_{Y},}$ which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology ${\displaystyle b\left(B(X,Y),X{\widehat {\otimes }}_{\pi }Y\right),}$ and in (Grothendieck 1955), Alexander Grothendieck was interested in when these two topologies were identical. This question is equivalent to the questions: Given a bounded subset ${\displaystyle B\subseteq X{\widehat {\otimes }}_{\pi }Y,}$ do there exist bounded subsets ${\displaystyle B_{1}\subseteq X}$ and ${\displaystyle B_{2}\subseteq Y}$ such that ${\displaystyle B}$ is a subset of the closed convex hull of ${\displaystyle B_{1}\otimes B_{2}:=\{b_{1}\otimes b_{2}:b_{1}\in B_{1},b_{2}\in B_{2}\}}$?

Grothendieck proved that these topologies are equal when ${\displaystyle X}$ and ${\displaystyle Y}$ are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck^[16]). They are also equal when both spaces are Fréchet with one of them being nuclear.^[12]

Strong dual and bidual[]

Given a locally convex TVS ${\displaystyle X,}$ ${\displaystyle X^{\prime }}$ is assumed to have the strong topology (so ${\displaystyle X^{\prime }=X_{b}^{\prime }}$) and unless stated otherwise, the same is true of the bidual ${\displaystyle X^{\prime \prime }}$ (so ${\displaystyle X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }.}$ Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Properties[]

• ${\displaystyle X\otimes _{\pi }Y}$ is Hausdorff if and only if both ${\displaystyle X}$ and ${\displaystyle Y}$ are Hausdorff.^[7]
• Suppose that ${\displaystyle u:X_{1}\to Y_{1}}$ and ${\displaystyle v:X_{2}\to Y_{2}}$ are two linear maps between locally convex spaces. If both ${\displaystyle u}$ and ${\displaystyle v}$ are continuous then so is their tensor product ${\displaystyle u\otimes v:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}.}$^[18]
  • ${\displaystyle u\otimes v:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}}$ has a unique continuous extension to ${\displaystyle X_{1}{\widehat {\otimes }}_{\pi }X_{2}}$ denoted by ${\displaystyle u{\widehat {\otimes }}v:X_{1}{\widehat {\otimes }}_{\pi }X_{2}\to Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}.}$
  • If in addition both ${\displaystyle u}$ and ${\displaystyle v}$ are TVS-homomorphisms and the image of each map is dense in its codomain, then ${\displaystyle u{\widehat {\otimes }}_{\pi }v:X_{1}{\widehat {\otimes }}_{\pi }X_{2}\to Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}}$ is a homomorphism whose image is dense in ${\displaystyle Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}}$; if ${\displaystyle X_{1}}$ and ${\displaystyle Y_{1}}$ are both metrizable then this image is equal to all of ${\displaystyle Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}.}$^[18]
  • There are examples of ${\displaystyle u}$ and ${\displaystyle v}$ such that both ${\displaystyle u}$ and ${\displaystyle v}$ are surjective homomorphisms but ${\displaystyle u{\widehat {\otimes }}_{\pi }v:X_{1}{\widehat {\otimes }}_{\pi }X_{2}\to Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}}$ is not surjective.^[19]
  • There are examples of ${\displaystyle u}$ and ${\displaystyle v}$ such that both ${\displaystyle u}$ and ${\displaystyle v}$ are TVS-embeddings but ${\displaystyle u{\widehat {\otimes }}_{\pi }v:X_{1}{\widehat {\otimes }}_{\pi }X_{2}\to Y_{1}{\widehat {\otimes }}_{\pi }Y_{2}}$ is not a TVS-embedding.^[19] In order for ${\displaystyle u{\widehat {\otimes }}_{\pi }v}$ to be a TVS-embedding, it is necessary and sufficient to additionally show that every equicontinuous subset of ${\displaystyle B\left(X_{1},X_{2}\right)}$ is the image under ${\displaystyle {}^{t}\left(u{\widehat {\otimes }}_{\pi }v\right)}$ of an equicontinuous subset of ${\displaystyle B\left(Y_{1},Y_{2}\right).}$^[19]
  • If all four spaces are normed then ${\displaystyle \|u\otimes v\|_{\pi }=\|u\|\|v\|.}$^[20]
• The π-topology is finer than the ε-topology (since the canonical bilinear map ${\displaystyle X\times Y\to X\otimes _{\epsilon }Y}$ is continuous).^[10]
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are Frechet spaces then ${\displaystyle X\otimes _{\pi }Y}$ is barelled.^[21]
• If ${\displaystyle Y}$ and ${\displaystyle \left(X_{\alpha }\right)}$ are locally convex spaces then the canonical map ${\displaystyle \left(\prod _{\alpha }X_{\alpha }\right){\widehat {\otimes }}_{\pi }Y\to \prod _{\alpha }\left(X_{\alpha }{\widehat {\otimes }}_{\pi }Y\right)}$ is a TVS-isomorphism.^[21]
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are Frechet spaces and ${\displaystyle Z}$ is a complete Hausdorff locally convex space, then the canonical vector space isomorphism ${\displaystyle I:B(X,Y;Z)\to L\left(X{\widehat {\otimes }}_{\pi }Y;Z\right)}$ becomes a homeomorphism when these spaces are given the topologies of uniform convergence on products of compact sets and, for the second one, the topology of compact convergence (i.e. ${\displaystyle I:B_{c}(X,Y;Z)\to L_{c}\left(X{\widehat {\otimes }}_{\pi }Y;Z\right)}$ is a TVS-isomorphism).^[22]
• Suppose ${\displaystyle X}$ and ${\displaystyle Y}$ are Frechet spaces. Every compact subset of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ is contained in the closed convex balanced hull of the tensor product if a compact subset of ${\displaystyle X}$ and a compact subset of ${\displaystyle Y.}$^[22]
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are nuclear then ${\displaystyle X\otimes _{\pi }Y}$ and ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ are nuclear.^[23]

Projective norm[]

Suppose now that ${\displaystyle \left(X,\|\,\cdot \,\|\right)}$ and ${\displaystyle \left(Y,\|\,\cdot \,\|\right)}$ are normed spaces. Then ${\displaystyle X\otimes _{\pi }Y}$ is a normable space with a canonical norm denoted by ${\displaystyle \|\,\cdot \,\|_{\pi }.}$ The ${\displaystyle \pi }$-norm is defined on ${\displaystyle X\otimes Y}$ by

where ${\displaystyle W}$ is the balanced convex hull of ${\displaystyle C_{p}\otimes C_{q}=\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}.}$ Given ${\displaystyle b}$ in ${\displaystyle X\otimes Y,}$ this can also be expressed as^[6] where the infimum is taken over all finite sequences ${\displaystyle x_{1},\ldots ,x_{n}\in X}$ and ${\displaystyle y_{1},\ldots ,y_{n}\in Y}$ (of the same length) such that ${\displaystyle b=x_{1}\otimes y_{1}+\cdots +x_{n}\otimes y_{n}.}$ If ${\displaystyle b}$ is in ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ then where the infimum is taken over all (finite or infinite) sequences ${\displaystyle x_{1},\ldots ,\in X}$ and ${\displaystyle y_{1},\ldots ,\in Y}$ (of the same length) such that ${\displaystyle b=x_{1}\otimes y_{1}+\cdots .}$^[24] Also, where the infimum is taken over all sequences ${\displaystyle \left(x_{i}\right)}$ in ${\displaystyle X}$ and ${\displaystyle \left(y_{i}\right)}$ in ${\displaystyle Y}$ and scalars ${\displaystyle \lambda _{1},\cdots }$ (of the same length) such that ${\displaystyle b=\lambda _{1}x_{1}\otimes y_{1}+\cdots ,}$ ${\displaystyle \|x_{i}\|=\|y_{i}\|=1,}$ and ${\displaystyle \sum _{i}|\lambda _{i}|<\infty .}$^[24] Also, where the infimum is taken over all sequences ${\displaystyle \left(x_{i}\right)}$ in ${\displaystyle X}$ and ${\displaystyle \left(y_{i}\right)}$ in ${\displaystyle Y}$ and scalars ${\displaystyle \lambda _{1},\cdots }$ (of the same length) such that ${\displaystyle b=\lambda _{1}x_{1}\otimes y_{1}+\cdots ,}$ ${\displaystyle \left(x_{i}\right)}$ and ${\displaystyle \left(y_{i}\right)}$ converge to the origin, and ${\displaystyle \sum _{i}|\lambda _{i}|<\infty .}$^[24]

If ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces then the closed unit ball of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ is the closed convex hull of the tensor product of the closed unit ball in ${\displaystyle X}$ with that of ${\displaystyle Y.}$^[25]

Properties[]

• For all normed spaces ${\displaystyle \left(Z,\|\,\cdot \,\|\right),}$ the canonical vector space isomorphism of ${\displaystyle B(X,Y;Z)}$ onto ${\displaystyle L\left(X\otimes _{\pi }Y;Z\right)}$ is an isometry.^[26]
• Suppose that ${\displaystyle \|\,\cdot \,\|}$ is a norm on ${\displaystyle X\otimes Y}$ and let the TVS topology that it induces on ${\displaystyle X\otimes Y}$ be denoted by ${\displaystyle \alpha .}$ If the canonical linear map of ${\displaystyle B(X,Y)}$ into ${\displaystyle \left(X\otimes Y\right)^{\#},}$ which is the algebraic dual of ${\displaystyle X\otimes Y,}$ is an isometry of ${\displaystyle B(X,Y)}$ onto ${\displaystyle \left(X\otimes _{\alpha }Y\right)^{\prime },}$ then ${\displaystyle \|\,\cdot \,\|=\|\,\cdot \,\|_{\pi }.}$^[26]

Preserved properties[]

• In general, the projective tensor product does not respect subspaces (e.g. if ${\displaystyle Z}$ is a vector subspace of ${\displaystyle X}$ then the TVS ${\displaystyle Z\otimes _{\pi }Y}$ has in general a coarser topology than the subspace topology inherited from ${\displaystyle X\otimes _{\pi }Y}$).^[27]
• Suppose that ${\displaystyle E}$ and ${\displaystyle F}$ are complemented subspaces of ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively. Then ${\displaystyle E\otimes F}$ is a complemented subvector space of ${\displaystyle X\otimes _{\pi }Y}$ and the projective norm on ${\displaystyle E\otimes _{\pi }F}$ is equivalent to the projective norm on ${\displaystyle X\otimes _{\pi }Y}$ restricted to the subspace ${\displaystyle E\otimes F}$; Furthermore, if ${\displaystyle X}$ and ${\displaystyle F}$ are complemented by projections of norm 1, then ${\displaystyle E\otimes _{\pi }F}$ is complemented by a projection of norm 1.^[27]
• If ${\displaystyle I:X\otimes _{\pi }Y\to Z}$ is an isometric embedding into a Banach space ${\displaystyle Z,}$ then its unique continuous extension ${\displaystyle I:X{\widehat {\otimes }}_{\pi }Y\to Z}$ is also an isometric embedding.
• If ${\displaystyle \alpha :W\to X}$ and ${\displaystyle \beta :Y\to Z}$ are quotient operators between Banach spaces, then so is ${\displaystyle \alpha {\widehat {\otimes }}_{\pi }\beta :W{\widehat {\otimes }}_{\pi }Y\to X{\widehat {\otimes }}_{\pi }Z.}$^[28]
  • A continuous linear operator ${\displaystyle \beta :Y\to Z}$ between normed spaces is a quotient operator if it is surjective and it maps the open unit ball of ${\displaystyle Y}$ into the open unit ball of ${\displaystyle Z,}$ or equivalently if for all ${\displaystyle z\in Z,}$ ${\displaystyle \|z\|=\inf _{y\in \beta ^{-1}(z)}\|y\|.}$ ^[28]
• Let ${\displaystyle X}$ and ${\displaystyle F}$ be vector subspaces of the Banach spaces ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively. Then ${\displaystyle E{\widehat {\otimes }}_{\pi }F}$ is a TVS-subspace of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ if and only if every bounded bilinear form on ${\displaystyle E\times F}$ extends to a continuous bilinear form on ${\displaystyle X\times Y}$ with the same norm.^[29]

Trace form[]

Suppose that ${\displaystyle X}$ is a locally convex spaces. There is a bilinear form on ${\displaystyle X\times X^{\prime }}$ defined by ${\displaystyle \left(x,x^{\prime }\right)\mapsto x^{\prime }(x),}$ which when ${\displaystyle X}$ is a Banach space has norm equal to 1. This bilinear form corresponds to a linear form on ${\displaystyle X\otimes X^{\prime }}$ given by mapping ${\displaystyle z:=\sum _{i=1}^{n}x_{i}\otimes x_{i}^{\prime }}$ to ${\displaystyle \sum _{i=1}^{n}x_{i}^{\prime }\left(x_{i}\right)}$ (where of course this value is in fact independent of the representation ${\displaystyle \sum _{i=1}^{n}x_{i}\otimes x_{i}^{\prime }}$ of ${\displaystyle z}$ chosen). Letting ${\displaystyle X^{\prime }}$ have its strong dual topology, we can continuously extend this linear map to a map ${\displaystyle \operatorname {Tr} :X{\widehat {\otimes }}_{\pi }X_{b}^{\prime }\to \mathbb {C} }$ (assuming that the vector spaces have scalar field ${\displaystyle \mathbb {C} }$) called the trace of ${\displaystyle X.}$ This name originates from the fact that if we write ${\displaystyle z=\sum _{i,j=1}^{n}z_{ij}e_{i}\otimes e_{j}^{\prime }}$ where ${\displaystyle e_{j}^{\prime }\left(e_{i}\right)=1}$ if ${\displaystyle i=j}$ and 0 otherwise, then ${\displaystyle \operatorname {Tr} (z)=\sum _{i=1}^{n}z_{ii}.}$^[30]

Duality with L(X; Y')[]

Assuming that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces over the field ${\displaystyle \mathbb {F} ,}$ one may define a dual system between ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ and ${\displaystyle L_{b}\left(X;Y^{\prime }\right)}$ with the duality map ${\displaystyle \left\langle \cdot ,\cdot \right\rangle :L_{b}\left(X;Y^{\prime }\right)\times \left(X{\widehat {\otimes }}_{\pi }Y\right)\to \mathbb {F} }$ defined by ${\displaystyle \langle u,z\rangle :=\operatorname {Tr} \left(\left(u\,{\widehat {\otimes }}_{\pi }\operatorname {Id} _{Y}\right)(z)\right),}$ where ${\displaystyle \operatorname {Id} _{Y}:Y\to Y}$ is the identity map and ${\displaystyle u\,{\widehat {\otimes }}_{\pi }\operatorname {Id} _{Y}:X{\widehat {\otimes }}_{\pi }Y\to Y^{\prime }{\widehat {\otimes }}_{\pi }Y}$ is the unique continuous extension of the continuous map ${\displaystyle u\,\otimes _{\pi }\operatorname {Id} _{Y}:X\otimes _{\pi }Y\to Y^{\prime }\otimes _{\pi }Y.}$ If we write ${\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}}$ with ${\displaystyle \sum _{i=1}^{\infty }\left|\lambda _{i}\right|<\infty }$ and the sequences ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ and ${\displaystyle \left(y_{i}\right)_{i=1}^{\infty }}$ each converging to zero, then we have

^[31]

Nuclear operators[]

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

where it can be shown that this value is independent of the representation of ${\displaystyle z}$ chosen.

Nuclear operators between Banach spaces[]

Assuming that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces, then the map ${\displaystyle I:X_{b}^{\prime }\otimes _{\pi }Y\to L_{b}(X;Y)}$ has norm ${\displaystyle 1}$ so it has a continuous extension to a map ${\displaystyle {\hat {I}}:X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y\to L_{b}(X;Y),}$ where it is known that this map is not necessarily injective.^[32] The range of this map is denoted by ${\displaystyle L^{1}(X;Y)}$ and its elements are called nuclear operators.^[33] ${\displaystyle L^{1}(X;Y)}$ is TVS-isomorphic to ${\displaystyle \left(X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y\right)/\operatorname {ker} {\hat {I}}}$ and the norm on this quotient space, when transferred to elements of ${\displaystyle L^{1}(X;Y)}$ via the induced map ${\displaystyle {\hat {I}}:\left(X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y\right)/\operatorname {ker} {\hat {I}}\to L^{1}(X;Y),}$ is called the trace-norm and is denoted by ${\displaystyle \|\,\cdot \,\|_{\operatorname {Tr} }.}$

Nuclear operators between locally convex spaces[]

Suppose that ${\displaystyle U}$ is a convex balanced closed neighborhood of the origin in ${\displaystyle X}$ and ${\displaystyle B}$ is a convex balanced bounded Banach disk in ${\displaystyle Y}$ with both ${\displaystyle X}$ and ${\displaystyle Y}$ locally convex spaces. Let ${\displaystyle p_{U}(x)=\inf _{r>0,x\in rU}r}$ and let ${\displaystyle \pi :X\to X/p_{U}^{-1}(0)}$ be the canonical projection. One can define the auxiliary 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 ${\displaystyle F_{B}=\operatorname {span} B}$ normed by ${\displaystyle p_{B}(y)=\inf _{r>0,y\in rB}r}$ 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 ${\displaystyle L\left({\hat {X}}_{U};Y_{B}\right)\to L(X;Y)}$ and we henceforth use this map to identify ${\displaystyle L\left({\hat {X}}_{U};Y_{B}\right)}$ as a subspace of ${\displaystyle L(X;Y).}$^[33]

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Hausdorff locally convex spaces. The union of all ${\displaystyle L^{1}\left({\hat {X}}_{U};Y_{B}\right)}$ as ${\displaystyle U}$ ranges over all closed convex balanced neighborhoods of the origin in ${\displaystyle X}$ and ${\displaystyle B}$ ranges over all bounded Banach disks in ${\displaystyle Y,}$ is denoted by ${\displaystyle L^{1}(X;Y)}$ and its elements are call nuclear mappings of ${\displaystyle X}$ into ${\displaystyle Y.}$^[33]

When ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces, then this new definition of nuclear mapping is consistent with the original one given for the special case where ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces.

Nuclear operators between Hilbert spaces[]

Every nuclear operator is an integral operator but the converse is not necessarily true. However, every integral operator between Hilbert spaces is nuclear.^[34]

Nuclear bilinear forms[]

There is a canonical vector space embedding ${\displaystyle J:X^{\prime }\otimes Y^{\prime }\to {\mathcal {B}}(X,Y)}$ defined by sending ${\displaystyle z:=\sum _{i}^{n}x_{i}^{\prime }\otimes y_{i}^{\prime }}$ to the map

where it can be shown that this value is independent of the representation of ${\displaystyle z}$ chosen.

Nuclear bilinear forms on Banach spaces[]

Assuming that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces, then the map ${\displaystyle J:X_{b}^{\prime }\otimes _{\pi }Y_{b}^{\prime }\to {\mathcal {B}}_{b}(X,Y)}$ has norm ${\displaystyle 1}$ so it has a continuous extension to a map ${\displaystyle {\hat {J}}:X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }\to {\mathcal {B}}_{b}(X,Y).}$ The range of this map is denoted by ${\displaystyle B^{1}(X,Y)}$ and its elements are called nuclear bilinear forms.^[36] ${\displaystyle B^{1}(X,Y)}$ is TVS-isomorphic to ${\displaystyle \left(X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }\right)/\operatorname {ker} {\hat {J}}}$ and the norm on this quotient space, when transferred to elements of ${\displaystyle B^{1}(X,Y)}$ via the induced map ${\displaystyle {\hat {J}}:\left(X_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }\right)/\operatorname {ker} {\hat {J}}\to B^{1}(X,Y),}$ is called the nuclear-norm and is denoted by ${\displaystyle \|\,\cdot \,\|_{\operatorname {N} }.}$

Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces and that ${\displaystyle N}$ is a continuous bilinear from on ${\displaystyle X\times Y.}$

• The following are equivalent:

1. ${\displaystyle N}$ is nuclear.
2. There exist bounded sequences ${\displaystyle \left(x_{i}^{\prime }\right)_{i=1}^{\infty }}$ in ${\displaystyle X_{b}^{\prime }}$ and ${\displaystyle \left(y_{i}^{\prime }\right)_{i=1}^{\infty }}$ in ${\displaystyle Y_{b}^{\prime }}$ such that ${\displaystyle \sum _{i=1}^{\infty }\|x_{i}^{\prime }\|\|y_{i}^{\prime }\|<\infty }$ and ${\displaystyle N}$ is equal to the mapping:^[37] ${\displaystyle N(x,y)=\sum _{i=1}^{\infty }x_{i}^{\prime }(x)y_{i}^{\prime }(y)}$ for all ${\displaystyle (x,y)\in X\times Y.}$

• In this case we call ${\displaystyle \sum _{i=1}^{\infty }x_{i}^{\prime }\otimes y_{i}^{\prime }}$ a nuclear representation of ${\displaystyle N.}$^[37]

The nuclear norm of ${\displaystyle N}$ is:

^[37] Note that ${\displaystyle \|N\|\leq \|N\|_{\operatorname {N} }.}$^[37]

Examples[]

Space of absolutely summable families[]

Throughout this section we fix some arbitrary (possibly uncountable) set ${\displaystyle A,}$ a TVS ${\displaystyle X,}$ and we let ${\displaystyle {\mathcal {F}}(A)}$ be the directed set of all finite subsets of ${\displaystyle A}$ directed by inclusion ${\displaystyle \subseteq .}$

Let ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ be a family of elements in a TVS ${\displaystyle X}$ and for every finite subset ${\displaystyle H}$ of ${\displaystyle A,}$ let ${\displaystyle x_{H}:=\sum _{i\in H}x_{i}.}$ We call ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ summable in ${\displaystyle X}$ if the limit ${\displaystyle \lim _{H\in {\mathcal {F}}(A)}x_{H}}$ of the net ${\displaystyle \left(x_{H}\right)_{H\in {\mathcal {F}}(A)}}$ converges in ${\displaystyle X}$ to some element (any such element is called its sum). We call ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ absolutely summable if it is summable and if for every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X,}$ the family ${\displaystyle \left(p\left(x_{\alpha }\right)\right)_{\alpha \in A}}$ is summable in ${\displaystyle \mathbb {R} .}$^[38] The set of all such absolutely summable families is a vector subspace of ${\displaystyle X^{A}}$ denoted by ${\displaystyle S_{a}.}$

Note that if ${\displaystyle X}$ is a metrizable locally convex space then at most countably many terms in an absolutely summable family are non-0. A metrizable locally convex space is nuclear if and only if every summable sequence is absolutely summable.^[39] It follows that a normable space in which every summable sequence is absolutely summable, is necessarily finite dimensional.^[39]

We now define a topology on ${\displaystyle S_{a}}$ in a very natural way. This topology turns out to be the projective topology taken from ${\displaystyle \ell ^{1}(A){\widehat {\otimes }}_{\pi }X}$ and transferred to ${\displaystyle S_{a}}$ via a canonical vector space isomorphism (the obvious one). This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define (from scratch) a topology on such a tensor product is frequently equivalent to the projective or injective tensor product topology.

Let ${\displaystyle {\mathfrak {U}}}$ denote a base of convex balanced neighborhoods of the origin in ${\displaystyle X}$ and for each ${\displaystyle U\in {\mathfrak {U}},}$ let ${\displaystyle \mu _{U}:X\to \mathbb {R} }$ denote its Minkowski functional. For any such ${\displaystyle U}$ and any ${\displaystyle x=\left(x_{\alpha }\right)_{\alpha \in A}\in S_{a},}$ let ${\displaystyle p_{U}(x):=\sum _{\alpha \in A}\mu _{U}\left(x_{\alpha }\right)}$ where ${\displaystyle p_{U}}$ defines a seminorm on ${\displaystyle S_{a}.}$ The family of seminorms ${\displaystyle \{p_{U}:U\in {\mathfrak {U}}\}}$ generates a topology making ${\displaystyle S_{a}}$ into a locally convex space. The vector space ${\displaystyle S_{a}}$ endowed with this topology will be denoted by ${\displaystyle \ell ^{1}[A,X].}$^[38] The special case where ${\displaystyle X}$ is the scalar field will be denoted by ${\displaystyle \ell ^{1}[A].}$

There is a canonical embedding of vector spaces ${\displaystyle \ell ^{1}(A)\otimes X\to \ell ^{1}[A,E]}$ defined by linearizing the bilinear map ${\displaystyle \ell ^{1}(A)\times X\to \ell ^{1}[A,E]}$ defined by ${\displaystyle \left(\left(r_{\alpha }\right)_{\alpha \in A},x\right)\mapsto \left(r_{\alpha }x\right)_{\alpha \in A}.}$^[38]

See also[]

• Auxiliary normed spaces
• Initial topology – coarsest topology making certain functions continuous
• Inductive tensor product
• Injective tensor product
• Integral map
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Nuclear operator
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces
• Topological vector space – Vector space with a notion of nearness

References[]

Bibliography[]

• Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4440-3. OCLC 185095773.
• Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156.
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
• Grothendieck, Grothendieck (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• ; (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Nlend, H (1977). Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis. Amsterdam New York New York: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland. ISBN 0-7204-0712-5. OCLC 2798822.
• 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.
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
• Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184.
• Schaefer, Helmut H.; (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.

External links[]

• Nuclear space at ncatlab