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:{{sfn|Trèves|2006|p=438}}

: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

$(p \otimes q)(b) = \inf_{r > 0,\, b \in r W} r$

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 $Y$ .{{sfn|Trèves|2006|p=435}}{{sfn|Trèves|2006|p=438}}

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.{{sfn|Trèves|2006|p=437}}

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 $X \otimes_\pi Y$ ;{{sfn|Trèves|2006|p=437}} if $X$ and $Y$ are Fréchet spaces then $X \otimes_\pi Y$ is barelled.{{sfn|Trèves|2006|p=445}}
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.{{sfn|Trèves|2006|p=439}}
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 $X \otimes_\pi Y$ ).{{sfn|Ryan|2002|p=18}}
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.{{sfn|Ryan|2002|p=18}}
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.{{sfn|Ryan|2002|p=24}}

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 {{em|not}} complete{{sfn|Ryan|2002|p=43}}). 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 $B(X, Y)$ .{{sfn|Schaefer|Wolff|1999|p=173}}

= 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 $X.$ {{sfn|Schaefer|Wolff|1999|p=120}} 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 $X.$ {{sfn|Schaefer|Wolff|1999|p=120}}

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.{{sfn|Schaefer|Wolff|1999|p=94}}

{{math theorem|name=Theorem|note=|style=|math_statement=

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

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=459-460}}|note=|style=|math_statement=

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 {{harv|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{{sfn|Schaefer|Wolff|1999|p=154}}). They are also equal when both spaces are Fréchet with one of them being nuclear.{{sfn|Schaefer|Wolff|1999|p=173}}

= 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:

{{math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=175-176}}|note=Grothendieck|style=|math_statement=

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 $N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b$ ;
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 $N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b$ ) is a reflexive space;
Every separately continuous bilinear form on $N^{\prime}_b \times Y^{\prime}_b$ is continuous;
Let $L\left(X^{\prime}_b, Y\right)$ be the space of bounded linear maps from $X^{\prime}_b$ to $Y$ . Then, its strong dual can be identified with $N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b,$ so in particular if $Y$ is reflexive then so is $L_b\left(X^{\prime}_b, Y\right).$

}}

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 $L^1_E$ 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 $X\to\Reals$ , have integral $0$ with respect to $\mu$ . Then $L^1_E$ is isometrically isomorphic to $L^1 \widehat{\otimes}_\pi E$ .{{sfn|Schaefer|Wolff|1999|p=95}}

Citations

References

{{cite book|last=Ryan|first=Raymond|title=Introduction to tensor products of Banach spaces|publisher=Springer|location=London New York|year=2002|isbn=1-85233-437-1|oclc=48092184}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Trèves François Topological vector spaces, distributions and kernels}}

External links

[https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

Category:Functional analysis

Category:Topological tensor products