Projective tensor product


The strongest locally convex topological vector space topology on, the tensor product of two locally convex TVSs, making the canonical map continuous is called the projective topology or the π-topology. When X ⊗ Y is endowed with this topology then it is denoted by and called the projective tensor product of X and Y.

Preliminaries

Throughout let X,Y, and Z be topological vector spaces and be a linear map.
Let X and Y be vector spaces and let Bi be the space of all bilinear maps defined on and going into the underlying scalar field.
For every define a canonical bilinear form by with domain Bi by for every.
This induces a canonical map defined by, where denotes the algebraic dual of Bi.
If we denote the span of the range of ? by X ⊗ Y then X ⊗ Y together with ? forms a tensor product of X and Y. This gives us a canonical tensor product of X and Y.
If Z is any other vector space then the mapping given by is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of X ⊗ Y with the space of bilinear forms on. Moreover, if X and Y are locally convex topological vector spaces and if X ⊗ Y is given the ?-topology then for every locally convex TVS Z, this map restricts to a vector space isomorphism from the space of continuous linear mappings onto the space of continuous bilinear mappings. In particular, the continuous dual of X ⊗ Y can be canonically identified with the space B of continuous bilinear forms on ; furthermore, under this identification the equicontinuous subsets of B are the same as the equicontinuous subsets of ''.

The projective tensor product

Tensor product of seminorms

Throughout we will let X and Y be locally convex topological vector spaces.
If p is a seminorm on X then will be its closed unit ball.
If p is a seminorm on X and q is a seminorm on Y then we can define the tensor product of p and q to be the map p ⊗ q defined on X ⊗ Y by
where W is the balanced convex hull of.
Given b in X ⊗ Y, this can also be expressed as
where the infimum is taken over all finite sequences and such that .
If b = x⊗y then we have
The seminorm p ⊗ q is a norm if and only if both p and q are norms.
If the topology of X is given by the family of seminorms then is a locally convex space whose topology if given by the family of all possible tensor products of the two families.
In particular, if X and Y are seminormed spaces with seminorms p and q, respectively, then is a seminormable space whose topology is defined by the seminorm p ⊗ q. If ' and ' are normed spaces then is also a normed space, called the projective tensor product of ' and ', where the topology induced by p ⊗ q is the same as the π-topology.
If W is a convex subset of then W is a neighborhood of 0 in if and only if the preimage of W under the map is a neighborhood of 0; equivalent, if and only if there exist open subsets and such that this preimage contains. It follows that if and are neighborhood bases of the origin in X and Y, respectively, then the set of convex hulls of all possible set form a neighborhood basis of the origin in.

Universal property

If ? is a locally convex TVS topology on X ⊗ Y, then ? is equal to the π-topology if and only if it has the following property:
In particular, the continuous dual space of is canonically isomorphic to the space, the space of continuous bilinear forms on.

The π-topology

Note that the canonical vector space isomorphism preserves equicontinuous subsets. Since is canonically isomorphic to the continuous dual of, place on X ⊗ Y the topology of uniform convergence on equicontinuous subsets of ; this topology is identical to the π-topology.

Preserved properties

Let X and Y be locally convex TVSs.
In general, the space is not complete, even if both X and Y are complete. However, can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by, via a linear topological embedding. Explicitly, this means that there is a continuous linear injection whose image is dense in and that is a TVS-isomorphism onto its image. Using this map, is identified as a subspace of.
The continuous dual space of is the same as that of, namely the space of continuous bilinear forms.:
Any continuous map on can be extended to a unique continuous map on. In particular, if and are continuous linear maps between locally convex spaces then their tensor product, which is necessarily continuous, can be extended to a unique continuous linear function, which may also be denoted by if no ambiguity would arise.
Note that if X and Y are metrizable then so are and, where in particular will be an F-space.

Grothendieck's representation of elements of X \widehat{\otimes}_{\pi} Y

Recall that in a Hausdorff locally convex space X, a sequence in X is absolutely convergent if for every continuous seminorm p on X. We write if the sequence of partial sums 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's possible to make the representation of z independent of the sequences and.

Topology of bi-bounded convergence

Let and denote the families of all bounded subsets of X and Y, 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, and in, Alexander Grothendieck was interested in when these two topologies were identical.
This question is equivalent to the questions: Given a bounded subset, do there exist bounded subsets and such that B is a subset of the closed convex hull of ?
Grothendieck proved that these topologies are equal when X and Y 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

Given a locally convex TVS X, is assumed to have the strong topology and unless stated otherwise, the same is true of the bidual : Let N and Y be locally convex TVSs with N nuclear. Assume that both N and Y are Fréchet spaces or else that they are both DF-spaces. Then:
  1. The strong dual of can be identified with ;
  2. The dibual of can be identified with ;
  3. If in addition Y is reflexive then is a reflexive space;
  4. Every separately continuous bilinear form on is continuous;
  5. The strong dual of can be identified with, so in particular if Y is reflexive then so is.

    Properties

Suppose now that and are normed spaces. Then is a normable space with a canonical norm denoted by.
The π-norm is defined on X ⊗ Y by
where W is the balanced convex hull of.
Given b in X ⊗ Y, this can also be expressed as
where the infimum is taken over all finite sequences and such that.
If b is in then
where the infimum is taken over all sequences and such that. Also,
where the infimum is taken over all sequences in X and in Y and scalars such that,, and. Also,
where the infimum is taken over all sequences in X and in Y and scalars such that, and converge to the origin, and.
If X and Y are Banach spaces then the closed unit ball of is the closed convex hull of the tensor product of the closed unit ball in X with that of Y.

Properties

Suppose that X is a locally convex spaces. There is a bilinear form on defined by, which when X is a Banach space has norm equal to 1. This bilinear form corresponds to a linear form on given by mapping to . Letting have its strong dual topology, we can continuously extend this linear map to a map called the trace of X.
This name originates from the fact that if we write where if i = j and 0 otherwise, then.

Duality with L(X; Y')

Assuming that X and Y are Banach spaces over the field, one may define a dual system between and with the duality map
defined by
,
where is the identity map and
is the unique continuous extension of the continuous map
.
If we write with and the sequences and each converging to zero, then we have

Nuclear operators

There is a canonical vector space embedding defined by sending to the map
where it can be shown that this value is independent of the representation of z chosen.

Nuclear operators between Banach spaces

Assuming that X and Y are Banach spaces, then the map has norm so it has a continuous extension to a map, where it is known that this map is not necessarily injective. The range of this map is denoted by and its elements are called nuclear operators. is TVS-isomorphic to and the norm on this quotient space, when transferred to elements of via the induced map, is called the trace-norm and is denoted by.

Nuclear operators between locally convex spaces

Suppose that U is a convex balanced closed neighborhood of the origin in X and B is a convex balanced bounded Banach disk in Y with both X and Y locally convex spaces. Let and let be the canonical projection. One can define the auxiliary Banach space with the canonical map whose image,, is dense in as well as the auxiliary space normed by and with a canonical map being the canonical injection.
Given any continuous linear map one obtains through composition the continuous linear map ; thus we have an injection and we henceforth use this map to identify as a subspace of.
When X and Y are Banach spaces, then this new definition of nuclear mapping is consistent with the original one given for the special case where X and Y are Banach spaces.

Nuclear operators between Hilbert spaces

Every nuclear operator is an integral operator but the converse is not necessarily true. However, every integral operator between Hilbert spaces is nuclear.
Theorem: Let X and Y be Hilbert spaces and endow with the trace-norm. When the space of compact linear operators is equipped with the operator norm then its dual is and its bidual is the space of all continuous linear operators.

Nuclear bilinear forms

There is a canonical vector space embedding defined by sending to the map
where it can be shown that this value is independent of the representation of z chosen.

Nuclear bilinear forms on Banach spaces

Assuming that X and Y are Banach spaces, then the map has norm so it has a continuous extension to a map. The range of this map is denoted by and its elements are called nuclear bilinear forms. is TVS-isomorphic to and the norm on this quotient space, when transferred to elements of via the induced map, is called the nuclear-norm and is denoted by.
Suppose that X and Y are Banach spaces and that is a continuous bilinear from on.
  1. is nuclear.
  2. There exist bounded sequences in and in such that and is equal to the mapping: for all.
The nuclear norm of N is:
Note that.

Examples

Space of absolutely 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. We call absolutely summable if it is summable and if for every continuous seminorm p on X, the family is summable in. The set of all such absolutely summable families is a vector subspace of denoted by.
Note that if X is a metrizable locally convex space then at most countably many terms in an absolutely summable family are non-0.
A metrizable locally convex space is nuclear if and only if every summable sequence is absolutely summable. It follows that a normable space in which every summable sequence is absolutely summable, is necessarily finite dimensional.
We now define a topology on in a very natural way. This topology turns out to be the projective topology taken from and transferred to 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 projective or injective 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. The family of seminorms generates a topology making into a locally convex space. The vector space 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.