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 $Y$.

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 ${\mathrm{\Phi }}_Z$ 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 $Y$. 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 $X{\hat{\otimes }}_{\pi }Y$ denotes the completion of the projective tensor product of $X$ and $Y$.

• If $X$ and $Y$ are both Hausdorff then so is $X{\otimes }_{\pi }Y$; 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 $E\hat{\otimes }F$ 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.

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 $X{\hat{\otimes }}_{\pi }Y$.

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 $B(X,Y)$.

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 $X.$ 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 $X.$

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 $z\in X{\hat{\otimes }}_{\pi }Y.$ Then $z$ is the sum of an absolutely convergent series $$z=\sum _{i=1}^{\mathrm{\infty }}{\lambda }_i{x}_i\otimes y_i$$ where $\sum _{i=1}^{\mathrm{\infty }}|{\lambda }_i|<\mathrm{\infty },$ and ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ and ${\left(y_i\right)}_{i=1}^{\mathrm{\infty }}$ 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 ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ and ${\left(y_i\right)}_{i=1}^{\mathrm{\infty }}.$

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 ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ and ${\left(y_i\right)}_{i=1}^{\mathrm{\infty }}$ contained in $U$ and $V,$ respectively, converging to the origin such that for every $z\in K_0$ there exists some ${\left({\lambda }_i\right)}_{i=1}^{\mathrm{\infty }}\in K_1$ such that $$z=\sum _{i=1}^{\mathrm{\infty }}{\lambda }_i{x}_i\otimes y_i.$$

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 $X{\hat{\otimes }}_{\pi }Y$ 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 ${\mathfrak{𝔅}}_X\times {\mathfrak{𝔅}}_Y,$ 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 $B\subseteq X\hat{\otimes }Y,$ 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 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 $b$:

1. The strong dual of $N{\hat{\otimes }}_{\pi }Y$ can be identified with $N_b^{\prime }{\hat{\otimes }}_{\pi }{Y}_b^{\prime }$;
2. The bidual of $N{\hat{\otimes }}_{\pi }Y$ can be identified with $N{\hat{\otimes }}_{\pi }{Y}^{\prime \prime }$;
3. If $Y$ is reflexive then $N{\hat{\otimes }}_{\pi }Y$ (and hence $N_b^{\prime }{\hat{\otimes }}_{\pi }{Y}_b^{\prime }$) is a reflexive space;
4. Every separately continuous bilinear form on $N_b^{\prime }\times Y_b^{\prime }$ is continuous;
5. Let $L(X_b^{\prime },Y)$ be the space of bounded linear maps from $X_b^{\prime }$ to $Y$. Then, its strong dual can be identified with $N_b^{\prime }{\hat{\otimes }}_{\pi }{Y}_b^{\prime },$ so in particular if $Y$ is reflexive then so is $L_b(X_b^{\prime },Y).$

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 $X\to \mathbb{ℝ}$, have integral $0$ with respect to $\mu$. Then $L_E^1$ is isometrically isomorphic to $L^1{\hat{\otimes }}_{\pi }E$.

See also

• Inductive tensor product
• 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

References

• Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. 

Further reading

• 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. 
• Grothendieck, Grothendieck (1966) (in fr). Produits tensoriels topologiques et espaces nucléaires. Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788. 
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541. 
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158. 

External links

• Nuclear space at ncatlab