Projective tensor product

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

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

for all ${\displaystyle b}$ in ${\displaystyle X\otimes Y}$, where ${\displaystyle W}$ is the balanced convex hull of the set ${\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}}$. The projective topology on ${\displaystyle X\otimes Y}$ is generated by the collection of such tensor products of the seminorms on ${\displaystyle X}$ and ${\displaystyle Y}$.[2][1] When ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces, this definition applied to the norms on ${\displaystyle X}$ and ${\displaystyle Y}$ gives a norm, called the projective norm, on ${\displaystyle X\otimes Y}$ which generates the projective topology.[3]

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]

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

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

• 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]

• 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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• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

• 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.
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• Nuclear space at ncatlab