Injective tensor product

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In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued functions.

Contents

Preliminaries and notation

Throughout let and be topological vector spaces and be a linear map.

Notation for topologies

Definition

Throughout let and 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

Despite the fact that the tensor product is a purely algebraic construct (its definition does not involve any topologies), the vector space of continuous bilinear functionals is nevertheless always a tensor product of and (that is, ) when is defined in the manner now described. [3]

For every let denote the bilinear form on defined by

This map is always continuous [3] and so the assignment that sends to the bilinear form induces a canonical map

whose image is contained in In fact, every continuous bilinear form on belongs to the span of this map's image (that is, ). The following theorem may be used to verify that together with the above map is a tensor product of and

Theorem  Let and be vector spaces and let be a bilinear map. Then is a tensor product of and if and only if [4] the image of spans all of (that is, ), and the vectors spaces and are -linearly disjoint, which by definition [5] means that for all sequences of elements and of the same finite length satisfying

  1. if all are linearly independent then all are and
  2. if all are linearly independent then all are

Equivalently, [4] and are -linearly disjoint if and only if for all linearly independent sequences in and all linearly independent sequences in the vectors are linearly independent.

Topology

Henceforth, all topological vector spaces considered will be assumed to be locally convex. If is any locally convex topological vector space, then [6] and for any equicontinuous subsets and and any neighborhood in define

where every set is bounded in [6] which is necessary and sufficient for the collection of all to form a locally convex TVS topology on [7] This topology is called the -topology and whenever a vector spaces is endowed with the -topology then this will be indicated by placing as a subscript before the opening parenthesis. For example, endowed with the -topology will be denoted by If is Hausdorff then so is the -topology. [6]

In the special case where is the underlying scalar field, is the tensor product and so the topological vector space is called the injective tensor product of and and it is denoted by This TVS is not necessarily complete so its completion, denoted by will be constructed. When all spaces are Hausdorff then is complete if and only if both and are complete, [8] in which case the completion of is a vector subspace of If and are normed spaces then so is where is a Banach space if and only if this is true of both and [9]

Equicontinuous sets

One reason for converging on equicontinuous subsets (of all possibilities) is the following important fact:

A set of continuous linear functionals on a TVS [note 1] is equicontinuous if and only if it is contained in the polar of some neighborhood of the origin in ; that is,

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 locally convex 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 [10]

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 any locally convex space 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: [11]
    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
  • An equicontinuous set is bounded in the topology of bounded convergence (that is, bounded in ). [11] 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:
    1. is equicontinuous;
    2. is bounded in the topology of pointwise convergence (that is, bounded in );
    3. is bounded in the topology of bounded convergence (that is, 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. [12]

  • If is a Baire space then any subset that is bounded in is necessarily equicontinuous. [12]
  • If is separable, is metrizable, and is a dense subset of then the topology of pointwise convergence on makes metrizable so that in particular, the subspace topology that any equicontinuous subset inherits from is metrizable. [11]

For equicontinuous subsets of the continuous dual space (where is now the underlying scalar field of ), the following hold:

  • The weak closure of an equicontinuous set of linear functionals on is a compact subspace of [11]
  • If is separable then every weakly closed equicontinuous subset of is a metrizable compact space when it is given the weak topology (that is, the subspace topology inherited from ). [11]
  • If is a normable space then a subset is equicontinuous if and only if it is strongly bounded (that is, bounded in ). [11]
  • If is a barrelled space then for any subset the following are equivalent: [12]
    1. is equicontinuous;
    2. is relatively compact in the weak dual topology;
    3. is weakly bounded;
    4. 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. [13]

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 has its original topology. [14]

There also exists a canonical vector space isomorphism [14]

To define it, for every separately continuous bilinear form defined on and every let be defined by

Because is canonically vector space-isomorphic to (via the canonical map value at ), will be identified as an element of 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 the canonical map becomes a TVS-isomorphism [14]

In particular, can be canonically TVS-embedded into ; furthermore the image in of under the canonical map consists exactly of the space of continuous linear maps whose image is finite dimensional. [9]

The inclusion always holds. If is normed then is in fact a topological vector subspace of And if in addition is Banach then so is (even if is not complete). [9]

Properties

The canonical map is always continuous [15] and the ε-topology is always coarser than the π-topology, [16] which is in turn coarser than the inductive topology (the finest locally convex TVS topology making separately continuous). The space is Hausdorff if and only if both and are Hausdorff. [15]

If and are normed then is normable in which case for all [17]

Suppose that and are two linear maps between locally convex spaces. If both and are continuous then so is their tensor product [18] Moreover:

Relation to projective tensor product and nuclear spaces

The projective topology or the -topology is the finest locally convex topology on that makes continuous the canonical map defined by sending to the bilinear form When is endowed with this topology then it will be denoted by and called the projective tensor product of and

The following definition was used by Grothendieck to define nuclear spaces. [22]

Definition 0: Let be a locally convex topological vector space. Then is nuclear if for any locally convex space 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

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 (by definition of the π-topology) so there exists a unique continuous linear extension

If and are Hilbert spaces then is injective and the dual of is canonically isometrically isomorphic to the vector space of nuclear operators from into (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 The map

is continuous and when is complete, it has a continuous extension

When and 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 into (which is a closed vector subspace of Hence is identical to space of compact operators from into (note the prime on ). The space of compact linear operators between any two Banach spaces (which includes Hilbert spaces) and is a closed subset of [23]

Furthermore, the canonical map is injective when and are Hilbert spaces. [23]

Integral forms and operators

Integral bilinear forms

Denote the identity map by

and let

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 [24] [25]   The dual of consists of exactly those continuous bilinear forms v on that can be represented in the form of a map

where and are some closed, equicontinuous subsets of and respectively, and is a positive Radon measure on the compact set with total mass Furthermore, if is an equicontinuous subset of 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. [26] An integral map is of the form, for every and

for suitable weakly closed and equicontinuous subsets 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

Examples

Space of summable families

Throughout this section we fix some arbitrary (possibly uncountable) set a TVS and we let be the directed set of all finite subsets of directed by inclusion

Let be a family of elements in a TVS and for every finite subset let We call summable in if the limit of the net converges in 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 in a very natural way. This topology turns out to be the injective topology taken from and transferred to 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 and for each let denote its Minkowski functional. For any such 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 [27] The special case where is the scalar field will be denoted by

There is a canonical embedding of vector spaces defined by linearizing the bilinear map defined by [27]

Theorem: [27]   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 then the continuous extension of this embedding is an isomorphism of TVSs. So in particular, if 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 [28] Suppose and is a function such that with a limit point of Say that is differentiable at if there exist vectors in called the partial derivatives of , such that

where

One may naturally extend the notion of continuously differentiable function to -valued functions defined on For any let denote the vector space of all -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 [29]   If 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 is a normed space and if is a compact set, then the -norm on is equal to [29] If and are two compact spaces, then where this canonical map is an isomorphism of Banach spaces. [29]

Spaces of sequences converging to 0

If is a normed space, then let denote the space of all sequences in that converge to the origin and give this space the norm Let denote Then for any Banach space is canonically isometrically isomorphic to [29]

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 and in variables, is a bounded subset of To generalize the topology of the Schwartz space to we give the topology of uniform convergence over of the functions as and vary over all possible pairs of polynomials in variables. [29]

Theorem [29]   If is a complete locally convex space, then is canonically isomorphic to

See also

Notes

  1. This is true even if is not assumed to be Hausdorff or locally convex.

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References

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Bibliography