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
Although the question of whether or not one vector space is a tensor product of two other vector spaces is a purely algebraic one (that is, the answer does not depend on the topologies of X or Y). 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 a bilinear form, denoted by the symbol x ⊗ y, from into the underlying field (i.e. ) 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:
- (Z, T) is a tensor product of X and Y;
- (a) the image of T spans all of Z, and (b) X and Y are T-linearly disjoint (this means that for all positive integers n and all elements and such that , (i) if all are linearly independent then all are 0, and (ii) if all are linearly independent then all are 0).
Equivalently, X and Y are T-linearly disjoint if and only if for all linearly independent sequences in X and all linearly independent sequences in Y, the vectors are linearly independent.
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
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.
always hold and whenever any one of these vector spaces is endowed with the ε-topology then this will be indicated by placing ε as a subscript before the opening parenthesis. For example, endowed with the ε-topology will be denoted by
In particular, when Z is the underlying scalar field then since the topological vector space will be denoted by which is called the injective tensor product of X and Y. This TVS is not necessarily complete so its completion will be denoted 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, denoted by 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.
One reason for converging on equicontinuous subsets (of all possibilities) is the following important fact:
- A set of continuous linear functionals H on a TVS (not necessarily Hausdorff or locally convex) is equicontinuous if and only if it is contained in the polar of some neighborhood U of in (i.e. ).
A TVS's topology is completely determined by the open neighborhoods of the origin. This fact together with the bipolar theorem means that via the operation of taking the polar of a subset, the collection of all equicontinuous subsets of "encodes" all information about 's given topology. Specifically, distinct TVS topologies on produce distinct collections of equicontinuous subsets and conversely, given any such collection of equicontinuous sets, the TVS's original topology can be recovered by taking the polar of every (equicontinuous) set in the collection. Thus through this identification, uniform convergence on the collection of equicontinuous subsets is essentially uniform convergence on the very topology of the TVS; this allows one to directly relate the injective topology with the given topologies of and
Furthermore, the topology of a locally convex Hausdorff space is identical to the topology of uniform convergence on the equicontinuous subsets of
For this reason, the article now lists some properties of equicontinuous sets that are relevant for dealing with the injective tensor product. Throughout and are arbitrary TVSs and is a collection of linear maps from into
- If is equicontinuous then the subspace topologies that inherits from the following topologies on are identical:
- the topology of precompact convergence;
- the topology of compact convergence;
- the topology of pointwise convergence;
- the topology of pointwise convergence on a given dense subset of .
- An equicontinuous set is bounded in the topology of bounded convergence (i.e. bounded in ). So in particular, will also bounded in every TVS topology that is coarser than the topology of bounded convergence.
- If is a barrelled space and is locally convex then for any subset the following are equivalent:
- is equicontinuous;
- is bounded in the topology of pointwise convergence (i.e. bounded in );
- is bounded in the topology of bounded convergence (i.e. bounded in ).
In particular, to show that a set is equicontinuous it suffices to show that it is bounded in the topology of pointwise converge.
- If is a Baire space then any subset that is bounded in is necessarily equicontinuous.
- If is separable, is metrizable, and is a dense subset of X, then the topology of pointwise convergence on makes metrizable so that in particular, the subspace topology that any equicontinuous subset inherits from is metrizable.
We now restrict our attention to properties of equicontinuous subsets of the continuous dual space (where Y is now the underlying scalar field of ).
- The weak closure of an equicontinuous set of linear functionals on is a compact subspace of .
- If is separable then every weakly closed equicontinuous subset of is a metrizable compact space when it is given the weak topology (i.e. the subspace topology inherited from ).
- If is a normable space then a subset is equicontinuous if and only if it is strongly bounded (i.e. bounded in ).
- If is a barrelled space then for any subset the following are equivalent:
- is equicontinuous;
- is relatively compact in the weak dual topology;
- is weakly bounded;
- is strongly bounded.
We mention some additional important basic properties relevant to the injective tensor product:
- Suppose that is a bilinear map where is a Fréchet space, is metrizable, and is locally convex. If is separately continuous then it is continuous.
Canonical identification of separately continuous bilinear maps with linear maps
The set equality always holds; that is, if is a linear map, then is continuous if and only if is continuous, where here Y has its original topology.
There also exists a canonical vector space isomorphism
To define it, for every separately continuous bilinear form defined on and every , let be defined by
Because is canonically vector space-isomorphic to Y (via the canonical map value at y), will be identified as an element of Y, which will be denoted 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.
The inclusion always holds. If X is normed then is in fact a topological vector subspace of . And if in addition Y is Banach then so is (even if X is not complete).
Relation to projective tensor product and nuclear spaces
The strongest locally convex topology on making the canonical map (defined by sending to the bilinear form ) 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 (particularly those that relate to nuclear operators and nuclear spaces).
Dual spaces of the injective tensor product and its completion
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 (by definition of the π-topology) 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 (with the trace norm).
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 (when the spaces are given their usual norms) whose range is the space of all compact linear operators from X into Y (which is a closed vector subspace of . Hence is identical to space of compact operators from into Y (note the prime on X). The space of compact linear operators between any two Banach spaces (which includes Hilbert 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
Denote the identity map by
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 (bilinear) forms on . The following theorem justifies the word integral.
Theorem The dual J(X, Y) 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(X, Y) 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.
Space of summable families
Throughout this section we fix some arbitrary (possibly uncountable) 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 (any such element is called its sum). 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 (the obvious one). 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 (from scratch) 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 .
- Theorem: The canonical embedding (of vector spaces) becomes an embedding of topological vector spaces when is given the injective topology and furthermore, its range is dense in its codomain. If is a completion of X then the continuous extension of this embedding is an isomorphism of TVSs. So in particular, if X is complete then is canonically isomorphic to .
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 (TVS).
- Definition Suppose and is a function such that with a limit point of . Then we say that f is differentiable at if there exist n vectors in Y, called the partial derivatives of f, such that
- in Y
- where .
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 (see the article: Differentiable vector-valued functions from Euclidean space).
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 .