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:Template:Sfn

For any locally convex topological vector space $Z$, if ${\mathrm{\Phi }}_Z$ 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 ${\mathrm{\Phi }}_Z$ 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 $$(p\otimes q)(b)=\underset{r>0,b\in rW}{inf}r$$ for all $b$ in $X\otimes Y$, where $W$ is the balanced convex hull of the set $\{x\otimes y:p(x)\le 1,q(y)\le 1\}$. The projective topology on $X\otimes Y$ is generated by the collection of such tensor products of the seminorms on $X$ and $Y$.Template:SfnTemplate:Sfn 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.Template:Sfn

Properties

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

• If $X$ and $Y$ are both Hausdorff then so is $X{\otimes }_{\pi }Y$;Template:Sfn if $X$ and $Y$ are Fréchet spaces then $X{\otimes }_{\pi }Y$ is barelled.Template:Sfn
• 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.Template:Sfn
• 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 $X{\otimes }_{\pi }Y$).Template:Sfn
• 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.Template:Sfn
• Let $E$ and $F$ be vector subspaces of the Banach spaces $X$ and $Y$, respectively. Then $E\hat{\otimes }F$ is a TVS-subspace of $X{\hat{\otimes }}_{\pi }Y$ 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.Template:Sfn

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 Template:Em completeTemplate:Sfn). 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 $X{\hat{\otimes }}_{\pi }Y$.

The continuous dual space of $X{\hat{\otimes }}_{\pi }Y$ is the same as that of $X{\otimes }_{\pi }Y$, namely, the space of continuous bilinear forms $B(X,Y)$.Template:Sfn

Grothendieck's representation of elements in the completion

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

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

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

Topology of bi-bounded convergence

Let ${𝔅}_X$ and ${𝔅}_Y$ denote the families of all bounded subsets of $X$ and $Y,$ respectively. Since the continuous dual space of $X{\hat{\otimes }}_{\pi }Y$ 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 ${𝔅}_X\times {𝔅}_Y,$ which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on $B(X,Y)$, and in Template:Harv, Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset $B\subseteq X\hat{\otimes }Y,$ 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 GrothendieckTemplate:Sfn). They are also equal when both spaces are Fréchet with one of them being nuclear.Template:Sfn

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:

Examples

• For $(X,𝒜,\mu )$ 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 $X\to ℝ$, have integral $0$ with respect to $\mu$. Then $L_E^1$ is isometrically isomorphic to $L^1{\hat{\otimes }}_{\pi }E$.Template:Sfn

See also

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References

• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels

Further reading

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• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
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External links

• Nuclear space at ncatlab