Injective tensor product


In mathematics, the injective tensor product of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.

Preliminaries and notation

Throughout let X,Y, and Z be topological vector spaces and be a linear map.
Throughout let X and Y be topological vector spaces with continuous dual spaces and. Note that almost all results described are independent of whether these vector spaces are over or but to simplify the exposition we will assume that they are over the field.

Continuous bilinear maps as a tensor product

Note that the question of whether or not one vector space is a tensor product of two other vector spaces is a purely algebraic one. Nevertheless, the vector space of continuous bilinear functionals is always a tensor product of X and Y, as we now describe.
For every we now define bilinear form, denoted by the symbol x ⊗ y, from into the underlying field by
This induces a canonical map defined by sending to the bilinear form.
The span of the range of this map is.
The following theorem may be used to verify that together with the above map ⊗ is a tensor product of X and Y.
Theorem Let X, Y, and Z be vector spaces and let be a bilinear map. Then the following are equivalent:
  1. is a tensor product of X and Y;
  2. the image of T spans all of Z, and X and Y are T-linearly disjoint if all are linearly independent then all are 0, and.

    Topology

Henceforth, all topological vector spaces considered will be assumed to be locally convex.
If Z is any locally convex topological vector space, then for any equicontinuous subsets and, and any neighborhood in Z, define
It can be shown that every set is bounded, which is necessary and sufficient for the collection of all such to form a locally convex TVS topology on called the ε-topology. When is endowed with the ε-topology, it will be denoted by .
.
In particular, when Z is the underlying scalar field then since, we will denote by, which is called the injective tensor product of X and Y. The space is not necessarily complete so we denote its completion by. The space is complete if and only if both X and Y are complete, in which case the completion of is a subvector space of.
If X and Y are normed then so is. And is a Banach space if and only if both X and Y are Banach spaces.

Equicontinuous sets

One reason for converging on equicontinuous subsets is the following important fact:
Since a TVS's topology is completely determined by the open neighborhoods of the origin, when combined with the bipolar theorem, this means that via the operation of taking the polar of a set, the collection of equicontinuous subsets of "encodes" all information about X's topology: distinct TVS topologies on X produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS's original topology by taking the polars of sets in the collection. Thus uniform convergence on the collection of equicontinuous subsets is essentially convergence on the very topologies of the TVSs and allows one to more directly relate this topology with the topology of the original spaces.
Furthermore, the topology of a locally convex Hausdorff space X is identical to the topology of uniform convergence on the equicontinuous subsets of.
For this reason, we now list some properties of equicontinuous sets that are relevant for dealing with the injective tensor product. Throughout X and Y are arbitrary TVSs and H is a collection of linear maps from X into Y.

  1. the topology of precompact convergence
  2. the topology of compact convergence
  3. the topology of pointwise convergence
  4. the topology of pointwise convergence on a given dense subset of X

  1. H is equicontinuous;
  2. H is bounded in the topology of pointwise convergence ;
  3. H is bounded in the topology of bounded convergence.
In particular, to show that a set H is equicontinuous it suffices to show that it's bounded in the topology of pointwise converge.
We now restrict our attention to properties of equicontinuous subsets of the continuous dual space .

  1. H is equicontinuous;
  2. H is relatively compact in the weak dual topology.
  3. H is weakly bounded;
  4. H is strongly bounded;
We mention some additional important basic properties relevant to the injective tensor product:
One may show that the set equality holds.
There is also a canonical vector space isomorphism.
To define it, for every separately continuous bilinear form defined on and every, let be defined by.
Since is canonically vector space-isomorphic to Y, we will identify as an element of Y, which we'll denote by.
This defines a map given by and so the canonical isomorphism is of course defined by.
When is given the topology of uniform convergence on equicontinous subsets of X′, the canonical map becomes a TVS-isomorphism. In particular, can be canonically TVS-embedded into ; furthermore the image in of under the canonical map J consists exactly of the space of continuous linear maps whose image is finite dimensional.
One always has. If X is normed then is in fact a topological vector subspace of. And if in addition Y is Banach then so is .

Properties

The strongest locally convex topology on making the canonical map continuous is called the projective topology or the π-topology. When is endowed with this topology then it will be denoted by and called the projective tensor product of X and Y.
The following definition was used by Grothendieck to define nuclear spaces.
Definition 0: Let X be a locally convex topological vector space. Then X is nuclear if for any locally convex space Y, the canonical vector space embedding is an embedding of TVSs whose image is dense in the codomain.

Canonical identifications of bilinear and linear maps

In this section we describe canonical identifications between spaces of bilinear and linear maps. These identifications will be used to define important subspaces and topologies.

Dual spaces of the injective tensor product and its completion

Suppose that denotes the TVS-embedding of into its completion and let be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of as being identical to the continuous dual space of.
The identity map is continuous so there exists a unique continuous linear extension. If X and Y are Hilbert spaces then is injective and the dual of is canonically isometrically isomorphic to the vector space of nuclear operators from X into Y.

Injective tensor product of Hilbert spaces

There is a canonical map that sends to the linear map defined by, where it may be shown that the definition of does not depend on the particular choice of representation of z. The map is continuous and when is complete, it has a continuous extension.
When X and Y are Hilbert spaces then is a TVS-embedding and isometry whose range is the space of all compact linear operators from X into Y. The space of compact linear operators between any two Banach spaces X and Y is a closed subset of.
Furthermore, the canonical map is injective when X and Y are Hilbert spaces.

Integral forms and operators

Integral bilinear forms

Let denote the identity map and denote its transpose, which is a continuous injection. Recall that is canonically identified with, the space of continuous bilinear maps on. In this way, the continuous dual space of can be canonically identified as a subvector space of, denoted by. The elements of are called integral forms on. The following theorem justifies the word integral.
Theorem The dual J of consists of exactly those continuous bilinear forms v on that can be represented in the form of a map
where S and T are some closed, equicontinuous subsets of and, respectively, and is a positive Radon measure on the compact set with total mass.
Furthermore, if A is an equicontinuous subset of J then the elements can be represented with fixed and running through a norm bounded subset of the space of Radon measures on.

Integral linear operators

Given a linear map, one can define a canonical bilinear form, called the associated bilinear form on, by.
A continuous map is called integral if its associated bilinear form is an integral bilinear form. An integral map is of the form, for every and :
for suitable weakly closed and equicontinuous aubsets and of and, respectively, and some positive Radon measure of total mass.

Canonical map into ''L''(''X''; ''Y'')

There is a canonical map that sends to the linear map defined by, where it may be shown that the definition of does not depend on the particular choice of representation of z.

Examples

Space of summable families

Throughout this section we fix some arbitrary set A, a TVS X, and we let be the directed set of all finite subsets of A directed by inclusion.
Let be a family of elements in a TVS X and for every finite subset H of A, let. We call summable in X if the limit of the net converges in X to some element. The set of all such summable families is a vector subspace of denoted by.
We now define a topology on S in a very natural way. This topology turns out to be the injective topology taken from and transferred to S via a canonical vector space isomorphism. This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define a topology on such a tensor product is frequently equivalent to the injective or projective tensor product topology.
Let denote a base of convex balanced neighborhoods of 0 in X and for each, let denote its Minkowski functional. For any such U and any, let
where defines a seminorm on S. The family of seminorms generates a topology making S into a locally convex space. The vector space S endowed with this topology will be denoted by. The special case where X is the scalar field will be denoted by.
There is a canonical embedding of vector spaces defined by linearizing the bilinear map defined by.

Space of continuously differentiable vector-valued functions

Throughout, let be an open subset of, where is an integer and let be a locally convex topological vector space.
One may naturally extend the notion of continuously differentiable function to Y-valued functions defined on.
For any, let denote the vector space of all Y-valued maps defined on and let denote the vector subspace of consisting of all maps in that have compact support.
One may then define topologies on and in the same manner as the topologies on and are defined for the space of distributions and test functions.
All of this work in extending the definition of differentiability and various topologies turns out to be exactly equivalent to simply taking the completed injective tensor product:
Theorem If Y is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product.

Spaces of continuous maps from a compact space

If Y is a normed space and if K is a compact set, then the -norm on is equal to.
If H and K are two compact spaces, then, where this canonical map is an isomorphism of Banach spaces.

Spaces of sequences converging to 0

If Y is a normed space, then let denote the space of all sequences in Y that converge to the origin and give this space the norm.
Let denote.
Then for any Banach space Y, is canonically isometrically isomorphic to.

Schwartz space of functions

We will now generalize the Schwartz space to functions valued in a TVS.
Let be the space of all such that for all pairs of polynomials P and Q in n variables, is a bounded subset of Y.
To generalize the topology of the Schwartz space to, we give the topology of uniform convergence over of the functions, as P and Q vary over all possible pairs of polynomials in n variables.
Theorem: If Y is a complete locally convex space, then is canonically isomorphic to.