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 ${\displaystyle X}$ and ${\displaystyle Y}$, the projective topology, or π-topology, on ${\displaystyle X\otimes Y}$ is the strongest topology which makes ${\displaystyle X\otimes Y}$ a locally convex topological vector space such that the canonical map ${\displaystyle (x,y)\mapsto x\otimes y}$ (from ${\displaystyle X\times Y}$ to ${\displaystyle X\otimes Y}$) is continuous. When equipped with this topology, ${\displaystyle X\otimes Y}$ is denoted ${\displaystyle X\otimes _{\pi }Y}$ and called the projective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$.

Definitions

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally convex topological vector spaces. Their projective tensor product ${\displaystyle X\otimes _{\pi }Y}$ is the unique locally convex topological vector space with underlying vector space ${\displaystyle X\otimes Y}$ having the following universal property:^[1]

For any locally convex topological vector space ${\displaystyle Z}$, if ${\displaystyle \Phi _{Z))$ is the canonical map from the vector space of bilinear maps ${\displaystyle X\times Y\to Z}$ to the vector space of linear maps ${\displaystyle X\otimes Y\to Z}$; then the image of the restriction of ${\displaystyle \Phi _{Z))$ to the continuous bilinear maps is the space of continuous linear maps ${\displaystyle X\otimes _{\pi }Y\to Z}$.

When the topologies of ${\displaystyle X}$ and ${\displaystyle Y}$ are induced by seminorms, the topology of ${\displaystyle X\otimes _{\pi }Y}$ is induced by seminorms constructed from those on ${\displaystyle X}$ and ${\displaystyle Y}$ as follows. If ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$, and ${\displaystyle q}$ is a seminorm on ${\displaystyle Y}$, define their tensor product ${\displaystyle p\otimes q}$ to be the seminorm on ${\displaystyle X\otimes Y}$ given by

$${\displaystyle (p\otimes q)(b)=\inf _{r>0,\,b\in rW}r}$$

${\displaystyle b}$

${\displaystyle X\otimes Y}$

${\displaystyle W}$

${\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\))$

${\displaystyle X\otimes Y}$

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle X\otimes Y}$

Properties

Throughout, all spaces are assumed to be locally convex. The symbol ${\displaystyle X{\widehat {\otimes ))_{\pi }Y}$ denotes the completion of the projective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$.

• If ${\displaystyle X}$ and ${\displaystyle Y}$ are both Hausdorff then so is ${\displaystyle X\otimes _{\pi }Y}$;^[3] if ${\displaystyle X}$ and ${\displaystyle Y}$ are Fréchet spaces then ${\displaystyle X\otimes _{\pi }Y}$ is barelled.^[4]
• For any two continuous linear operators ${\displaystyle u_{1}:X_{1}\to Y_{1))$ and ${\displaystyle u_{2}:X_{2}\to Y_{2))$, their tensor product (as linear maps) ${\displaystyle u_{1}\otimes u_{2}:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2))$ is continuous.^[5]
• 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 ${\displaystyle X\otimes _{\pi }Y}$).^[6]
• If ${\displaystyle E}$ and ${\displaystyle F}$ are complemented subspaces of ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively, then ${\displaystyle E\otimes F}$ is a complemented vector subspace 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 ${\displaystyle E\otimes F}$ is complemented by a projection of norm 1.^[6]
• Let ${\displaystyle E}$ and ${\displaystyle F}$ be vector subspaces of the Banach spaces ${\displaystyle X}$ and ${\displaystyle Y}$, respectively. Then ${\displaystyle E{\widehat {\otimes ))F}$ 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.^[7]

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^[8]). 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 ${\displaystyle X{\widehat {\otimes ))_{\pi }Y}$.

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 ${\displaystyle B(X,Y)}$.^[9]

Grothendieck's representation of elements in the completion

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 ${\displaystyle X.}$^[10] 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 ${\displaystyle X.}$^[10]

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.^[11]

Theorem — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be metrizable locally convex TVSs and let ${\displaystyle z\in X{\widehat {\otimes ))_{\pi }Y.}$ Then ${\displaystyle z}$ is the sum of an absolutely convergent series

$${\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i))$$

where ${\displaystyle \sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,}$ and ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty ))$ and ${\displaystyle \left(y_{i}\right)_{i=1}^{\infty ))$ are null sequences in ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively.

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 ${\displaystyle \left(y_{i}\right)_{i=1}^{\infty }.}$

Theorem^[12] — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and let ${\displaystyle U}$ (resp. ${\displaystyle V}$) be a balanced open neighborhood of the origin in ${\displaystyle X}$ (resp. in ${\displaystyle Y}$). Let ${\displaystyle K_{0))$ be a compact subset of the convex balanced hull of ${\displaystyle U\otimes V:=\{u\otimes v:u\in U,v\in V\}.}$ There exists a compact subset ${\displaystyle K_{1))$ of the unit ball in ${\displaystyle \ell ^{1))$ and sequences ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty ))$ and ${\displaystyle \left(y_{i}\right)_{i=1}^{\infty ))$ contained in ${\displaystyle U}$ and ${\displaystyle V,}$ respectively, converging to the origin such that for every ${\displaystyle z\in K_{0))$ there exists some ${\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1))$ such that

$${\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.}$$

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 on ${\displaystyle B(X,Y)}$, 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 ${\displaystyle B\subseteq X{\widehat {\otimes ))Y,}$ 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^[13]). They are also equal when both spaces are Fréchet with one of them being nuclear.^[9]

Strong dual and bidual

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

Examples

• For ${\displaystyle (X,{\mathcal {A)),\mu )}$ a measure space, let ${\displaystyle L^{1))$ be the real Lebesgue space ${\displaystyle L^{1}(\mu )}$; let ${\displaystyle E}$ be a real Banach space. Let ${\displaystyle L_{E}^{1))$ be the completion of the space of simple functions ${\displaystyle X\to E}$, modulo the subspace of functions ${\displaystyle X\to E}$ whose pointwise norms, considered as functions ${\displaystyle X\to \mathbb {R} }$, have integral ${\displaystyle 0}$ with respect to ${\displaystyle \mu }$. Then ${\displaystyle L_{E}^{1))$ is isometrically isomorphic to ${\displaystyle L^{1}{\widehat {\otimes ))_{\pi }E}$.^[15]