# 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 denoted $X\otimes _{\pi }Y$ and called the projective tensor product of $X$ and $Y$. It is a particular instance of a topological tensor product.

## Definitions

Let $X$ and $Y$ be locally convex topological vector spaces. Their projective tensor product $X\otimes _{\pi }Y$ is the unique locally convex topological vector space with underlying vector space $X\otimes Y$ having the following universal property:
    For any locally convex topological vector space $Z$, if $\Phi _{Z}$ is the canonical map from the vector space of bilinear maps $X\times Y\to Z$ to the vector space of linear maps $X\otimes Y\to Z$, then the image of the restriction of $\Phi _{Z}$ to the continuous bilinear maps is the space of continuous linear maps $X\otimes _{\pi }Y\to Z$.
When the topologies of $X$ and $Y$ are induced by seminorms, the topology of $X\otimes _{\pi }Y$ is induced by seminorms constructed from those on $X$ and $Y$ as follows. If $p$ is a seminorm on $X$, and $q$ is a seminorm on $Y$, define their tensor product $p\otimes q$ to be the seminorm on $X\otimes Y$ given by $(p\otimes q)(b)=\inf _{r>0,\,b\in rW}r$ for all $b$ in $X\otimes Y$, where $W$ is the balanced convex hull of the set $\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}$. The projective topology on $X\otimes Y$ is generated by the collection of such tensor products of the seminorms on $X$ and $Y$. When $X$ and $Y$ are normed spaces, this definition applied to the norms on $X$ and $Y$ gives a norm, called the projective norm, on $X\otimes Y$ which generates the projective topology.

## Properties

Throughout, all spaces are assumed to be locally convex. The symbol $X{\widehat {\otimes }}_{\pi }Y$ denotes the completion of the projective tensor product of $X$ and $Y$.
* If $X$ and $Y$ are both Hausdorff then so is $X\otimes _{\pi }Y$; if $X$ and $Y$ are Fréchet spaces then $X\otimes _{\pi }Y$ is barelled.
* For any two continuous linear operators $u_{1}:X_{1}\to Y_{1}$ and $u_{2}:X_{2}\to Y_{2}$, their tensor product (as linear maps) $u_{1}\otimes u_{2}:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}$ is continuous.
* In general, the projective tensor product does not respect subspaces (e.g. if $Z$ is a vector subspace of $X$ then the TVS $Z\otimes _{\pi }Y$ has in general a coarser topology than the subspace topology inherited from $X\otimes _{\pi }Y$).
* If $E$ and $F$ are complemented subspaces of $X$ and $Y,$ respectively, then $E\otimes F$ is a complemented vector subspace of $X\otimes _{\pi }Y$ and the projective norm on $E\otimes _{\pi }F$ is equivalent to the projective norm on $X\otimes _{\pi }Y$ 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.
* 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 $X{\widehat {\otimes }}_{\pi }Y$ 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.

## Completion

In general, the space $X\otimes _{\pi }Y$ is not complete, even if both $X$ and $Y$ are complete (in fact, if $X$ and $Y$ are both infinite-dimensional Banach spaces then $X\otimes _{\pi }Y$ is necessarily not complete). However, $X\otimes _{\pi }Y$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by $X{\widehat {\otimes }}_{\pi }Y$.
The continuous dual space of $X{\widehat {\otimes }}_{\pi }Y$ is the same as that of $X\otimes _{\pi }Y$, 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 $z\in X{\widehat {\otimes }}_{\pi }Y.$ 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 $X{\widehat {\otimes }}_{\pi }Y$ 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 $b$:
* The strong dual of $N{\widehat {\otimes }}_{\pi }Y$ can be identified with $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }$;
* The bidual of $N{\widehat {\otimes }}_{\pi }Y$ can be identified with $N{\widehat {\otimes }}_{\pi }Y^{\prime \prime }$;
* If $Y$ is reflexive then $N{\widehat {\otimes }}_{\pi }Y$ (and hence $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }$) is a reflexive space;
* Every separately continuous bilinear form on $N_{b}^{\prime }\times Y_{b}^{\prime }$ is continuous;
* 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 $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime },$ so in particular if $Y$ is reflexive then so is $L_{b}\left(X_{b}^{\prime },Y\right).$

## Examples

* For $(X,{\mathcal {A}},\mu )$ a measure space, let $L^{1}$ be the real Lebesgue space $L^{1}(\mu )$; let $E$ be a real Banach space. Let $L_{E}^{1}$ be the completion of the space of simple functions $X\to E$, modulo the subspace of functions $X\to E$ whose pointwise norms, considered as functions $X\to \mathbb {R}$, have integral $0$ with respect to $\mu$. Then $L_{E}^{1}$ is isometrically isomorphic to $L^{1}{\widehat {\otimes }}_{\pi }E$.