The
strongest locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) topology on
the tensor product of two locally convex TVSs, making the canonical map
(defined by sending
to
) continuous is called the projective topology or the π-topology. When
is endowed with this topology then it is denoted by
and called the projective tensor product of
and
Preliminaries
Throughout let
and
be
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s and
be a linear map.
*
is a topological homomorphism or homomorphism, if it is linear, continuous, and
is an
open map, where
the image of
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.
).
* 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 (e.g.
denotes an element of
and not, say, a derivative and the variables
and
need not be related in any way).
* A linear map
from a Hilbert space into itself is called positive if
for every
In this case, there is a unique positive map
called the square-root of
such that
** If
is any continuous linear map between Hilbert spaces, then
is always positive. Now let
denote its positive square-root, which is called the absolute value of
Define
first on
by setting
for
and extending
continuously to
and then define
on
by setting
for
and extend this map linearly to all of
The map
is a surjective isometry and
* A linear map
is called compact or completely continuous if there is a neighborhood
of the origin in
such that
is
precompact in
** In a Hilbert space, positive compact linear operators, say
have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:
::There is a sequence of positive numbers, decreasing and either finite or else converging to 0,
and a sequence of nonzero finite dimensional subspaces
of
(
) with the following properties: (1) the subspaces
are pairwise orthogonal; (2) for every
and every
; and (3) the orthogonal of the subspace spanned by
is equal to the kernel of
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.
** Every
induces a map
defined by
is the coarsest topology on
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
A canonical tensor product as a subspace of the dual of Bi(X, Y)
Let
and
be vector spaces (no topology is needed yet) and let
be the space of all
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
s defined on
and going into the underlying scalar field.
For every
define a canonical bilinear form by
with domain
by
for every
This induces a canonical map
defined by
where
denotes the
algebraic dual of
If we denote the span of the range of
by
then
together with
forms a
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of
and
(where
). This gives us a canonical tensor product of
and
If
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
with the space of bilinear forms on
Moreover, if
and
are locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs) and if
is given the
-topology then for every locally convex TVS
this map restricts to a vector space isomorphism
from the space of ''continuous'' linear mappings onto the space of bilinear mappings. In particular, the continuous dual of
can be canonically identified with the space
of continuous bilinear forms on
; furthermore, under this identification the
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
subsets of
are the same as the equicontinuous subsets of ''
The projective tensor product
Tensor product of seminorms
Throughout we will let
and
be
locally convex topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(local convexity allows us to define useful topologies).
If
is a seminorm on
then
will be its closed unit ball.
If
is a seminorm on
and
is a seminorm on
then we can define the tensor product of
and
to be the map
defined on
by
where
is the
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
convex hull of
Given
in
this can also be expressed as
where the infimum is taken over all finite sequences
and
(of the same length) such that
(recall that it may not be possible to express
as a
simple tensor).
If
then we have
The seminorm
is a norm if and only if both
and
are norms.
If the topology of
(resp.
) is given by the family of seminorms
(resp.
) then
is a locally convex space whose topology is given by the family of all possible tensor products of the two families (i.e. by
).
In particular, if
and
are seminormed spaces with seminorms
and
respectively, then
is a seminormable space whose topology is defined by the seminorm
If
and
are normed spaces then
is also a normed space, called the projective tensor product of
and
where the topology induced by
is the same as the π-topology.
If
is a convex subset of
then
is a neighborhood of 0 in
if and only if the preimage of
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
and
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
(
with this topology will be denoted by
), then
is equal to the π-topology if and only if it has the following property:
:For every locally convex TVS
if
is the canonical map from the space of all bilinear mappings of the form
going into the space of all linear mappings of
then when the domain of
is restricted to
then the range of this restriction is the space
of continuous linear operators
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
the topology of uniform convergence on equicontinuous subsets of
; this topology is identical to the π-topology.
Preserved properties
Let
and
be locally convex TVSs.
* If both
and
are Hausdorff (resp. locally convex,
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
,
semi-metrizable,
normable,
semi-normable) then so is
Completion
In general, the space
is not complete, even if both
and
are complete (in fact, if
and
are both infinite-dimensional Banach spaces then
is necessarily 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
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
. Explicitly, this means that there is a continuous linear
injection
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
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
and
are metrizable then so are
and
where in particular
will be an
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that
# Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
.
Grothendieck's representation of elements of
In a Hausdorff locally convex space
a sequence
in
is absolutely convergent if
for every continuous seminorm
on
We write
if the sequence of partial sums
converges to
in
The following fundamental result in the theory of topological tensor products is due to
Alexander Grothendieck.
The next theorem shows that it is possible to make the representation of
independent of the sequences
and
Topology of bi-bounded convergence
Let
and
denote the families of all bounded subsets of
and
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
is a subset of the closed convex hull of
?
Grothendieck proved that these topologies are equal when
and
are both Banach spaces or both are
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
s (a class of spaces introduced by Grothendieck). They are also equal when both spaces are Fréchet with one of them being nuclear.
Strong dual and bidual
Given a locally convex TVS
is assumed to have the strong topology (so
) and unless stated otherwise, the same is true of the bidual
(so
Alexander Grothendieck characterized the strong dual and bidual for certain situations:
Properties
*
is Hausdorff if and only if both
and
are Hausdorff.
* Suppose that
and
are two linear maps between locally convex spaces. If both
and
are continuous then so is their tensor product
**
has a unique continuous extension to
denoted by
** If in addition both
and
are TVS-homomorphisms and the image of each map is dense in its codomain, then
is a homomorphism whose image is dense in
; if
and
are both metrizable then this image is equal to all of
** There are examples of
and
such that both
and
are surjective homomorphisms but
is surjective.
** There are examples of
and
such that both
and
are TVS-embeddings but
is a TVS-embedding. In order for
to be a TVS-embedding, it is necessary and sufficient to additionally show that every equicontinuous subset of
is the image under
of an equicontinuous subset of
** If all four spaces are normed then
* The π-topology is finer than the
ε-topology (since the canonical bilinear map
is continuous).
* If
and
are Frechet spaces then
is barelled.
* If
and
are locally convex spaces then the canonical map
is a TVS-isomorphism.
* If
and
are Frechet spaces and
is a complete Hausdorff locally convex space, then the canonical vector space isomorphism
becomes a homeomorphism when these spaces are given the topologies of uniform convergence on products of compact sets and, for the second one, the topology of compact convergence (i.e.
is a TVS-isomorphism).
* Suppose
and
are Frechet spaces. Every compact subset of
is contained in the closed convex balanced hull of the tensor product if a compact subset of
and a compact subset of
* If
and
are nuclear then
and
are nuclear.
Projective norm
Suppose now that
and
are normed spaces. Then
is a normable space with a canonical norm denoted by
The
-norm is defined on
by
where
is the
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
convex hull of
Given
in
this can also be expressed as
where the infimum is taken over all finite sequences
and
(of the same length) such that
If
is in
then
where the infimum is taken over all (finite or infinite) sequences
and
(of the same length) such that
Also,
where the infimum is taken over all sequences
in
and
in
and scalars
(of the same length) such that
and
Also,
where the infimum is taken over all sequences
in
and
in
and scalars
(of the same length) such that
and
converge to the origin, and
If
and
are Banach spaces then the closed unit ball of
is the closed convex hull of the tensor product of the closed unit ball in
with that of
Properties
* For all normed spaces
the canonical vector space isomorphism of
onto
is an isometry.
* Suppose that
is a norm on
and let the TVS topology that it induces on
be denoted by
If the canonical linear map of
into
which is the algebraic dual of
is an isometry of
onto
then
Preserved properties
* In general, the projective tensor product does not respect subspaces (e.g. if
is a vector subspace of
then the TVS
has in general a coarser topology than the subspace topology inherited from
).
* Suppose that
and
are complemented subspaces of
and
respectively. Then
is a complemented subvector space of
and the projective norm on
is equivalent to the projective norm on
restricted to the subspace
; Furthermore, if
and
are complemented by projections of norm 1, then
is complemented by a projection of norm 1.
* If
is an isometric embedding into a Banach space
then its unique continuous extension
is also an isometric embedding.
* If
and
are quotient operators between Banach spaces, then so is
** A continuous linear operator
between normed spaces is a quotient operator if it is surjective and it maps the open unit ball of
into the open unit ball of
or equivalently if for all
* Let
and
be vector subspaces of the Banach spaces
and
respectively. Then
is a TVS-subspace of
if and only if every bounded bilinear form on
extends to a continuous bilinear form on
with the same norm.
Trace form
Suppose that
is a locally convex spaces. There is a bilinear form on
defined by
which when
is a Banach space has norm equal to 1. This bilinear form corresponds to a linear form on
given by mapping
to
(where of course this value is in fact independent of the representation
of
chosen). Letting
have its strong dual topology, we can continuously extend this linear map to a map
(assuming that the vector spaces have scalar field
) called the trace of
This name originates from the fact that if we write
where
if
and 0 otherwise, then
Duality with L(X; Y')
Assuming that
and
are Banach spaces over the field
one may define a
dual system
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb.
Duality theory, the study of dual ...
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
chosen.
Nuclear operators between Banach spaces
Assuming that
and
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
is a convex balanced closed neighborhood of the origin in
and
is a convex balanced bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
in
with both
and
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 (continuous) 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
Let
and
be Hausdorff locally convex spaces. The union of all
as
ranges over all closed convex balanced neighborhoods of the origin in
and
ranges over all bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
s in
is denoted by
and its elements are call nuclear mappings of
into
When
and
are Banach spaces, then this new definition of ''nuclear mapping'' is consistent with the original one given for the special case where
and
are Banach spaces.
Nuclear operators between Hilbert spaces
Every nuclear operator is an
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms invol ...
but the converse is not necessarily true. However, every integral operator between
Hilbert spaces is nuclear.
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
chosen.
Nuclear bilinear forms on Banach spaces
Assuming that
and
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
and
are Banach spaces and that
is a continuous bilinear from on
* The following are equivalent:
#
is nuclear.
# There exist bounded sequences
in
and
in
such that
and
is equal to the mapping:
for all
* In this case we call
a nuclear representation of
The nuclear norm of
is:
Note that
Examples
Space of absolutely summable families
Throughout this section we fix some arbitrary (possibly
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
) set
a TVS
and we let
be the
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
of all finite subsets of
directed by inclusion
Let
be a family of elements in a TVS
and for every finite subset
of
let
We call
summable in
if the limit
of the
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
converges in
to some element (any such element is called its sum). We call
absolutely summable if it is summable and if for every continuous seminorm
on
the family
is summable in
The set of all such absolutely summable families is a vector subspace of
denoted by
Note that if
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 (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 projective or
injective tensor product topology.
Let
denote a base of convex balanced neighborhoods of the origin 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