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 and, the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map is continuous. When equipped with this topology, is denoted and called the projective tensor product of and. It is a particular instance of a topological tensor product.

Definitions

Let and be locally convex topological vector spaces. Their projective tensor product is the unique locally convex topological vector space with underlying vector space having the following universal property:
When the topologies of and are induced by seminorms, the topology of is induced by seminorms constructed from those on and as follows. If is a seminorm on, and is a seminorm on, define their tensor product to be the seminorm on given by
for all in, where is the balanced convex hull of the set. The projective topology on is generated by the collection of such tensor products of the seminorms on and.
When and are normed spaces, this definition applied to the norms on and gives a norm, called the projective norm, on which generates the projective topology.

Properties

Throughout, all spaces are assumed to be locally convex. The symbol denotes the completion of the projective tensor product of and.
  • If and are both Hausdorff then so is ; if and are Fréchet spaces then is barelled.
  • For any two continuous linear operators and, their tensor product is continuous.
  • In general, the projective tensor product does not respect subspaces.
  • If and are complemented subspaces of and respectively, then is a complemented vector subspace of and the projective norm on is equivalent to the projective norm on restricted to the subspace. Furthermore, if and are complemented by projections of norm 1, then is complemented by a projection of norm 1.
  • Let and be vector subspaces of the Banach spaces and, respectively. Then is a TVS-subspace of if and only if every bounded bilinear form on extends to a continuous bilinear form on with the same norm.

Completion

In general, the space is not complete, even if both and are complete. However, can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by.
The continuous dual space of is the same as that of, namely, the space of continuous bilinear forms.

Grothendieck's representation of elements in the completion

In a Hausdorff locally convex space a sequence in is absolutely convergent if for every continuous seminorm on We write if the sequence of partial sums converges to in
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 independent of the sequences and

Topology of bi-bounded convergence

Let and denote the families of all bounded subsets of and respectively. Since the continuous dual space of is the space of continuous bilinear forms we can place on the topology of uniform convergence on sets in which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on, and in, Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset do there exist bounded subsets and such that is a subset of the closed convex hull of ?
Grothendieck proved that these topologies are equal when and are both Banach spaces or both are DF-spaces. They are also equal when both spaces are Fréchet with one of them being nuclear.

Strong dual and bidual

Let be a locally convex topological vector space and let be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Examples

  • For a measure space, let be the real Lebesgue space ; let be a real Banach space. Let be the completion of the space of simple functions, modulo the subspace of functions whose pointwise norms, considered as functions, have integral with respect to. Then is isometrically isomorphic to.