Projective tensor product

In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces $X$ and $Y$, the projective topology, or π-topology, on $X\otimes Y$ is the strongest topology which makes $X\otimes Y$ a locally convex topological vector space such that the canonical map $(x,y)\mapsto x\otimes y$ (from $X\times Y$ to $X\otimes Y$) is continuous. When equipped with this topology, $X\otimes Y$ is called the projective tensor product of $X$ and $Y$.

Properties

Throughout, all spaces are assumed to be locally convex topological vector spaces unless stated otherwise. When referring to a topological space, the symbol $X\otimes Y$ denotes the projective tensor product unless stated otherwise.

• $X\otimes Y$ is Hausdorff if and only if both $X$ and $Y$ are Hausdorff.
• If both $X$ and $Y$ are metrizable (resp. semi-metrizable, normable, semi-normable) then so is $X\otimes Y.$
• For any two continuous linear operators $u_{1}:X_{1}\to Y_{1}$ and $u_{2}:X_{2}\to Y_{2}$, their tensor product $u_{1}\otimes u_{2}:X_{1}\otimes X_{2}\to Y_{1}\otimes Y_{2}$ is continuous.
• When $X_{1},X_{2}$ are Banach spaces, each map $u\otimes v:X_{1}\otimes X_{2}\to Y_{1}\otimes Y_{2}$ has a unique continuous extension to the completion of $X_{1}\otimes X_{2}$.
• If $X$ and $Y$ are Frechet spaces then $X\otimes Y$ is barelled.
• For any $Y$ and collection $\left(X_{\alpha }\right)$, the canonical map $\left(\prod _{\alpha }X_{\alpha }\right){\widehat {\otimes }}Y\to \prod _{\alpha }\left(X_{\alpha }{\widehat {\otimes }}Y\right)$ is an isomorphism of topological vector spaces.
• If $X$ and $Y$ are nuclear then $X\otimes Y$ and $X{\widehat {\otimes }}Y$ are nuclear.
• In general, the projective tensor product does not respect subspaces (e.g. if $Z$ is a vector subspace of $X$ then the TVS $Z\otimes Y$ has in general a coarser topology than the subspace topology inherited from $X\otimes Y$).
• If $E$ and $F$ are complemented subspaces of $X$ and $Y,$ respectively, then $E\otimes F$ is a complemented vector subspace of $X\otimes Y$ and the projective norm on $E\otimes F$ is equivalent to the projective norm on $X\otimes Y$ restricted to the subspace $E\otimes F$. Furthermore, if $X$ and $F$ are complemented by projections of norm 1, then $E\otimes F$ is complemented by a projection of norm 1.
• If $I:X\otimes Y\to Z$ is an isometric embedding into a Banach space $Z,$ then its unique continuous extension $I:X{\widehat {\otimes }}Y\to Z$ is also an isometric embedding.
• Let $E$ and $F$ be vector subspaces of the Banach spaces $X$ and $Y,$ respectively. Then $E{\widehat {\otimes }}F$ is a TVS-subspace of $X{\widehat {\otimes }}Y$ if and only if every bounded bilinear form on $E\times F$ extends to a continuous bilinear form on $X\times Y$ with the same norm.

Universal property

For any locally convex topological vector space topology $\tau$ on $X\otimes Y$ ($X\otimes Y$ with this topology will be denoted by $X\otimes _{\tau }Y$), then $\tau$ is identical to the projective topology if and only if it has the following property:

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

In particular, the continuous dual space of the projective tensor product $X\otimes Y$ is canonically isomorphic to the space $B(X,Y)$ of continuous bilinear forms on $X\times Y$.

Normed spaces

Suppose now that $\left(X,\|\,\cdot \,\|\right)$ and $\left(Y,\|\,\cdot \,\|\right)$ are normed spaces. Then $X\otimes Y$ is a normable space with a canonical norm denoted by $\|\,\cdot \,\|_{\pi }.$ The $\pi$-norm is defined on $X\otimes Y$ by $$\|b\|_{\pi }:=\inf _{r>0,b\in rW}r$$ where $W$ is the balanced convex hull of $C_{p}\otimes C_{q}=\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}.$ Given $b$ in $X\otimes Y,$ this can also be expressed as $$\|b\|_{\pi }:=\inf \sum _{i}\|x_{i}\|\|y_{i}\|$$ where the infimum is taken over all finite sequences $x_{1},\ldots ,x_{n}\in X$ and $y_{1},\ldots ,y_{n}\in Y$ (of the same length) such that $b=x_{1}\otimes y_{1}+\cdots +x_{n}\otimes y_{n}.$ If $b$ is in $X{\widehat {\otimes }}Y$ then $$\|b\|_{\pi }:=\inf \sum _{i}\|x_{i}\|\|y_{i}\|$$ where the infimum is taken over all (finite or infinite) sequences $x_{1},\ldots ,\in X$ and $y_{1},\ldots ,\in Y$ (of the same length) such that $b=x_{1}\otimes y_{1}+\cdots .$ If $X$ and $Y$ are Banach spaces then the closed unit ball of $X{\widehat {\otimes }}Y$ is the closed convex hull of the tensor product of the closed unit ball in $X$ with that of $Y.$

Seminormed spaces

For $p$ a seminorm on $X$, let $C_{p}:=\{x\in X:p(x)\leq 1\}$ denote its closed unit ball.

If $p$ is a seminorm on $X$ and $q$ is a seminorm on $Y$, we define the tensor product of $p$ and $q$ to be the map $p\otimes q$ defined on $X\otimes Y$ by $$(p\otimes q)(b):=\inf _{r>0,b\in rW}r$$ where $W$ is the balanced convex hull of $C_{p}\otimes C_{q}=\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}.$ Given $b$ in $X\otimes Y,$ this can also be expressed as $$(p\otimes q)(b):=\inf \sum _{i}p(x_{i})q(y_{i})$$ where the infimum is taken over all finite sequences $x_{1},\ldots ,x_{n}\in X$ and $y_{1},\ldots ,y_{n}\in Y$ (of the same length) such that $b=x_{1}\otimes y_{1}+\cdots +x_{n}\otimes y_{n}$ (recall that it may not be possible to express $b$ as a simple tensor). If $b=x\otimes y$ then we have $$(p\otimes q)(x\otimes y)=p(x)q(y).$$ The seminorm $p\otimes q$ is a norm if and only if both $p$ and $q$ are norms.

If the topology of $X$ is given by the family of seminorms $\{p_{\alpha }\}$, and that of $Y$ by the seminorms $\{q_{\beta }\}$, then the topology on the vector space $X\otimes Y$ generated by the seminorms $p_{\alpha }\otimes q_{\beta }$ is identical to the projective topology. In particular, if $X$ and $Y$ are seminormed spaces with seminorms $p$ and $q,$ respectively, then $X\otimes _{\pi }Y$ is a seminormable space whose topology is defined by the seminorm $p\otimes q.$

Completion

In general, the space $X\otimes Y$ is not complete, even if both $X$ and $Y$ are complete (in fact, if $X$ and $Y$ are both infinite-dimensional Banach spaces then $X\otimes Y$ is necessarily not complete). However, $X\otimes Y$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by $X{\widehat {\otimes }}Y,$ via a linear embedding.

The continuous dual space of $X{\widehat {\otimes }}Y$ is the same as that of $X\otimes Y$, namely, the space of continuous bilinear forms $B(X,Y)$.

Grothendieck's representation of elements in the completion

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

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

Theorem—Let $X$ and $Y$ be metrizable locally convex TVSs and let $z\in X{\widehat {\otimes }}Y.$ Then $z$ is the sum of an absolutely convergent series $$z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}$$ where $\sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,$ and $\left(x_{i}\right)_{i=1}^{\infty }$ and $\left(y_{i}\right)_{i=1}^{\infty }$ are null sequences in $X$ and $Y,$ respectively.

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

Theorem—Let $X$ and $Y$ be Fréchet spaces and let $U$ (resp. $V$) be a balanced open neighborhood of the origin in $X$ (resp. in $Y$). Let $K_{0}$ be a compact subset of the convex balanced hull of $U\otimes V:=\{u\otimes v:u\in U,v\in V\}.$ There exists a compact subset $K_{1}$ of the unit ball in $\ell ^{1}$ and sequences $\left(x_{i}\right)_{i=1}^{\infty }$ and $\left(y_{i}\right)_{i=1}^{\infty }$ contained in $U$ and $V,$ respectively, converging to the origin such that for every $z\in K_{0}$ there exists some $\left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1}$ such that $$z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.$$

Topology of bi-bounded convergence

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

Grothendieck proved that these topologies are equal when $X$ and $Y$ are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck). They are also equal when both spaces are Fréchet with one of them being nuclear.

Strong dual and bidual

Let $X$ be a locally convex topological vector space and let $X^{\prime }$ be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Theorem (Grothendieck)—Let $N$ and $Y$ be locally convex topological vector spaces with $N$ nuclear. Assume that both $N$ and $Y$ are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted $b$:

1. The strong dual of $N{\widehat {\otimes }}Y$ can be identified with $N_{b}^{\prime }{\widehat {\otimes }}Y_{b}^{\prime }$;
2. The bidual of $N{\widehat {\otimes }}Y$ can be identified with $N{\widehat {\otimes }}Y^{\prime \prime }$;
3. If $Y$ is reflexive then $N{\widehat {\otimes }}Y$ (and hence $N_{b}^{\prime }{\widehat {\otimes }}Y_{b}^{\prime }$) is a reflexive space;
4. Every separately continuous bilinear form on $N_{b}^{\prime }\times Y_{b}^{\prime }$ is continuous;
5. Let $L\left(X_{b}^{\prime },Y\right)$ be the space of bounded linear maps from $X_{b}^{\prime }$ to $Y$. Then, its strong dual can be identified with $N_{b}^{\prime }{\widehat {\otimes }}Y_{b}^{\prime },$ so in particular if $Y$ is reflexive then so is $L_{b}\left(X_{b}^{\prime },Y\right).$

Examples

Space of absolutely summable families

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

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

Note that if $X$ 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. It follows that a normable space in which every summable sequence is absolutely summable, is necessarily finite dimensional.

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

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

Theorem—The canonical embedding (of vector spaces) $\ell ^{1}(A)\otimes X\to \ell ^{1}[A,E]$ becomes an embedding of topological vector spaces $\ell ^{1}(A)\otimes _{\pi }X\to \ell ^{1}[A,E]$ when $\ell ^{1}(A)\otimes X$ is given the projective topology and furthermore, its range is dense in its codomain. If ${\hat {X}}$ is a completion of $X$ then the continuous extension $\ell ^{1}(A){\widehat {\otimes }}_{\pi }X\to \ell ^{1}\left[A,{\hat {X}}\right]$ of this embedding $\ell ^{1}(A)\otimes _{\pi }X\to \ell ^{1}\left[A,X\right]\subseteq \ell ^{1}\left[A,{\hat {X}}\right]$ is an isomorphism of TVSs. So in particular, if $X$ is complete then $\ell ^{1}(A){\widehat {\otimes }}_{\pi }X$ is canonically isomorphic to $\ell ^{1}[A,E].$

See also

• Inductive tensor product – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback
• Injective 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

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External links

• Nuclear space at ncatlab