Projective tensor product explained

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 ⊗ Y is the strongest topology which makes X ⊗ Y a locally convex topological vector space such that the canonical map (x,y) \mapsto x ⊗ y (from X x Y to X ⊗ Y ) is continuous. When equipped with this topology, X ⊗ Y is denoted X ⊗ _{\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 ⊗ _{\pi} Y is the unique locally convex topological vector space with underlying vector space X ⊗ Y having the following universal property:

For any locally convex topological vector space

Z , if \Phi _{Z} is the canonical map from the vector space of bilinear maps X x Y \to Z to the vector space of linear maps X ⊗ 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 ⊗ _{\pi} Y \to Z .When the topologies of X and Y are induced by seminorms, the topology of X ⊗ _{\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 ⊗ q to be the seminorm on X ⊗ Y given by$$(p\; \backslash otimes\; q)(b)\; =\; \backslash inf\_\; r$$for all b in X ⊗ Y , where W is the balanced convex hull of the set \left\{ x ⊗ y : p(x) \leq 1 , q(y) \leq 1 \right\} . The projective topology on X ⊗ 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 ⊗ Y which generates the projective topology.

Properties

Throughout, all spaces are assumed to be locally convex. The symbol

X \widehat{ ⊗ } _{\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 ⊗ _{\pi} Y ; if X and Y are Fréchet spaces then X ⊗ _{\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} ⊗ u _{2} : X _{1} ⊗ _{\pi} X _{2} \to Y _{1} ⊗ _{\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 ⊗ _{\pi} Y has in general a coarser topology than the subspace topology inherited from X ⊗ _{\pi} Y ).

• If

E and F are complemented subspaces of X and Y, respectively, then E ⊗ F is a complemented vector subspace of X ⊗ _{\pi} Y and the projective norm on E ⊗ _{\pi} F is equivalent to the projective norm on X ⊗ _{\pi} Y restricted to the subspace E ⊗ F . Furthermore, if X and F are complemented by projections of norm 1, then E ⊗ 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 \widehat{ ⊗ } F is a TVS-subspace of X \widehat{ ⊗ } _{\pi} Y if and only if every bounded bilinear form on E x F extends to a continuous bilinear form on X x Y with the same norm.

Completion

In general, the space

X ⊗ _{\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 ⊗ _{\pi} Y is necessarily complete). However, X ⊗ _{\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{ ⊗ } _{\pi} Y .

The continuous dual space of

X \widehat{ ⊗ } _{\pi} Y is the same as that of X ⊗ _{\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)} ^{inf ty} ^{i =1} in X is absolutely convergent if \sum ^{inf ty} _{i =1} p \left(x _{i\right)} < inf ty for every continuous seminorm p on X. We write x = \sum ^{inf ty} _{i =1} x _{i} if the sequence of partial sums \left(\sum ^{n} _{i =1} x _{i\right)} ^{inf ty} ^{n =1} converges to x in X.

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

The next theorem shows that it is possible to make the representation of

z independent of the sequences \left(x _{i\right)} ^{inf ty} ^{i =1} and \left(y _{i\right)} ^{inf ty} ^{i =1} .

Topology of bi-bounded convergence

Let

ak{B} _{X} and ak{B} _{Y} denote the families of all bounded subsets of X and Y, respectively. Since the continuous dual space of X \widehat{ ⊗ } _{\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 ak{B} _{X} x ak{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, 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{ ⊗ } 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} ⊗ B _{2} : = \{ b _{1} ⊗ 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:

Examples

• For

(X, l{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{ ⊗ } _{\pi} E .

References

• Book: Ryan, Raymond. Introduction to tensor products of Banach spaces. Springer. London New York. 2002. 1-85233-437-1. 48092184.

Further reading

• Book: Diestel, Joe. The metric theory of tensor products : Grothendieck's résumé revisited. American Mathematical Society. Providence, R.I. 2008. 978-0-8218-4440-3. 185095773.
  • Book: Grothendieck, Grothendieck. Produits tensoriels topologiques et espaces nucléaires. American Mathematical Society. Providence. 1966. 0-8218-1216-5. 1315788. fr.
• Book: Pietsch, Albrecht. Nuclear locally convex spaces. Springer-Verlag. Berlin, New York. 1972. 0-387-05644-0. 539541.
• Book: Wong. Schwartz spaces, nuclear spaces, and tensor products. Springer-Verlag. Berlin New York. 1979. 3-540-09513-6. 5126158.

External links

• Nuclear space at ncatlab