Projective tensor product

Given locally convex topological vector spaces $X$ and $Y$, the projective topology, or π-topology, is the strongest topology on the vector space $X\otimes Y$ which makes it 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.

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 $Y.$

The projective tensor product

Tensor product of seminorms

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 $p\otimes q.$

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 $X\times Y$.

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 question is equivalent to the questions: 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

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

Theorem (Grothendieck)—Let $N$ and $Y$ be locally convex TVSs with $N$ nuclear. Assume that both $N$ and $Y$ are Fréchet spaces or else that they are both DF-spaces. Then:

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

Trace form

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

Duality with L(X; Y')

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

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

• Auxiliary normed spaces
• Initial topology – Coarsest topology making certain functions continuous
• Inductive tensor product – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback
• Injective tensor product
• Integral map – Mathematical functionPages displaying short descriptions of redirect targets
• Locally convex topological vector space – Vector space with a topology defined by convex open sets
• Nuclear operator – Linear operator related to topological vector spaces
• 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

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External links

• Nuclear space at ncatlab