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.
Preliminaries and notation
Throughout let
and
be topological vector spaces and
be a linear map.
is a topological homomorphism or homomorphism, if it is linear, continuous, and
is an open map, where
has the subspace topology induced by
- If
is a subspace of
then both the quotient map
and the canonical injection
are homomorphisms. In particular, any linear map
can be canonically decomposed as follows:
where
defines a bijection.
- The set of continuous linear maps
(resp. continuous bilinear maps
) will be denoted by
(resp.
) where if
is the scalar field then we may instead write
(resp.
). - The set of separately continuous bilinear maps
(that is, continuous in each variable when the other variable is fixed) will be denoted by
where if
is the scalar field then we may instead write ![{\displaystyle {\mathcal {B}}(X,Y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2775bf0ce5e43e9c30a0367e41efca693c25d0d2)
- We will denote the continuous dual space of
by
and the algebraic dual space (which is the vector space of all linear functionals on
whether continuous or not) by
- To increase the clarity of the exposition, we use the common convention of writing elements of
with a prime following the symbol (for example,
denotes an element of
and not, say, a derivative and the variables
and
need not be related in any way).
Notation for topologies
denotes the coarsest topology on
making every map in
continuous and
or
denotes
endowed with this topology.
denotes weak-* topology on
and
or
denotes
endowed with this topology. - Note that every
induces a map
defined by
is the coarsest topology on X′ making all such maps continuous.
denotes the topology of bounded convergence on
and
or
denotes
endowed with this topology.
denotes the topology of bounded convergence on
or the strong dual topology on
and
or
denotes
endowed with this topology. - As usual, if
is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be ![{\displaystyle b\left(X^{\prime },X\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0e55178a0f99c5872922f4637574702c1516d2)
denotes the Mackey topology on
or the topology of uniform convergence on the convex balanced weakly compact subsets of
and
or
denotes
endowed with this topology.
is the finest locally convex TVS topology on
whose continuous dual space is equal to ![{\displaystyle X^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/980210d5ccf78c678264901dc7b0ce8a53d827bc)
denotes the Mackey topology on
or the topology of uniform convergence on the convex balanced weakly compact subsets of
and
or
denotes
endowed with this topology. - Note that
![{\displaystyle \tau \left(X^{\prime },X\right)\subseteq b\left(X^{\prime },X\right)\subseteq \tau \left(X^{\prime },X^{\prime \prime }\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a71d417143162b344e1cd8750f69af82b64698f5)
denotes the topology of uniform convergence on equicontinuous subsets of
and
or
denotes
endowed with this topology. - If
is a set of linear mappings
then
is equicontinuous if and only if it is equicontinuous at the origin; that is, if and only if for every neighborhood
of the origin in
there exists a neighborhood
of the origin in
such that
for every ![{\displaystyle \lambda \in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b6531a5b3652f9187851378a8a983825c7a957)
- A set
of linear maps from
to
is called equicontinuous if for every neighborhood
of the origin in
there exists a neighborhood
of the origin in
such that
for all ![{\displaystyle h\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d10c356f8b2cea62359891c45c61a5670d8ade85)
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.
For every
let
denote the bilinear form on
defined by
This map
is always continuous 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 the image of
spans all of
(that is,
), and the vectors spaces
and
are
-linearly disjoint, which by definition means that for all sequences of elements
and
of the same finite length
satisfying
- if all
are linearly independent then all
are
and - if all
are linearly independent then all
are ![{\displaystyle 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962)
Equivalently,
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
and for any equicontinuous subsets
and
and any neighborhood
in
define
where every set
is bounded in
which is necessary and sufficient for the collection of all
to form a locally convex TVS topology on
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.
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, 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
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, ![{\displaystyle H\subseteq U^{\circ }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd74b1e558db7e5a9a925dc2c524e305a0e562e)
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
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: - 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
![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
- An equicontinuous set
is bounded in the topology of bounded convergence (that is, 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 (that is, bounded in
);
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.
- 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
then the topology of pointwise convergence on
makes
metrizable so that in particular, the subspace topology that any equicontinuous subset
inherits from
is metrizable.
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 ![{\displaystyle X_{\sigma }^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdf484c00434dfc521d9d7156326897dca1af10)
- 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
). - If
is a normable space then a subset
is equicontinuous if and only if it is strongly bounded (that is, 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
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
(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
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.
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).
Properties
The canonical map
is always continuous and the ε-topology is always coarser than the π-topology, 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.
If
and
are normed then
is normable in which case for all
Suppose that
and
are two linear maps between locally convex spaces. If both
and
are continuous then so is their tensor product
Moreover:
- If
and
are both TVS-embeddings then so is ![{\displaystyle u{\widehat {\otimes }}_{\varepsilon }v:X_{1}{\widehat {\otimes }}_{\varepsilon }X_{2}\to Y_{1}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f49b929f192f77bea93531e0b7cfc4a74a4ea1d5)
- If
(resp.
) is a linear subspace of
(resp.
) then
is canonically isomorphic to a linear subspace of
and
is canonically isomorphic to a linear subspace of ![{\displaystyle X_{2}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78a5f852461413d854a7726a79d3dacaf39ac398)
- There are examples of
and
such that both
and
are surjective homomorphisms but
is not a homomorphism. - If all four spaces are normed then
![{\displaystyle \|u\otimes v\|_{\varepsilon }=\|u\|\|v\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d70ab21fa51eadce193ddd6b9646b6037da6cf)
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.
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
Furthermore, the canonical map
is injective when
and
are Hilbert spaces.
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 — 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. 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
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
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
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 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:
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
If
and
are two compact spaces, then
where this canonical map is an isomorphism of Banach spaces.
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
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.
See also
Notes
- ^ This is true even if
is not assumed to be Hausdorff or locally convex.
References
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