In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, spaces of
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s between two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s can be endowed with a variety of
topologies
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The article
operator topologies In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space .
Introduction
Let (T_n)_ be a sequence of linear operators on the Banach spac ...
discusses topologies on spaces of linear maps between
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
s, whereas this article discusses topologies on such spaces in the more general setting of
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).
Topologies of uniform convergence on arbitrary spaces of maps
Throughout, the following is assumed:
- is any non-empty set and is a non-empty collection of subsets of
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
by subset inclusion (i.e. for any there exists some such that ).
- is a
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 ...
(not necessarily Hausdorff or locally convex).
- is a basis of neighborhoods of 0 in
- is a vector subspace of
[Because is just a set that is not yet assumed to be endowed with any vector space structure, should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined.] which denotes the set of all -valued functions with domain
đť’˘-topology
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
For any subsets
and
let
The family
forms a
neighborhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
at the origin for a unique translation-invariant topology on
where this topology is necessarily a vector topology (that is, it might not make
into a TVS).
This topology does not depend on the neighborhood basis
that was chosen and it is known as the topology of uniform convergence on the sets in
or as the
-topology.
However, this name is frequently changed according to the types of sets that make up
(e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details
[In practice, usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, is the collection of compact subsets of (and is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of ]).
A subset
of
is said to be fundamental with respect to
if each
is a subset of some element in
In this case, the collection
can be replaced by
without changing the topology on
One may also replace
with the collection of all subsets of all finite unions of elements of
without changing the resulting
-topology on
Call a subset
of
-bounded if
is a bounded subset of
for every
Properties
Properties of the basic open sets will now be described, so assume that
and
Then
is an
absorbing subset of
if and only if for all
absorbs
.
If
is
balanced (respectively,
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
) then so is
The equality
always holds.
If
is a scalar then
so that in particular,
Moreover,
and similarly
For any subsets
and any non-empty subsets
which implies:
- if then
- if then
- For any and subsets of if then
For any family
of subsets of
and any family
of neighborhoods of the origin in
Uniform structure
For any
and
be any
entourage
An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to:
Arts and entertainment
* L'entourage, French hip hop / rap collecti ...
of
(where
is endowed with its
canonical uniformity), let
Given
the family of all sets
as
ranges over any fundamental system of entourages of
forms a fundamental system of entourages for a uniform structure on
called or simply .
The is the least upper bound of all
-convergence uniform structures as
ranges over
Nets and uniform convergence
Let
and let
be a
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 ...
in
Then for any subset
of
say that
converges uniformly to
on
if for every
there exists some
such that for every
satisfying
(or equivalently,
for every
).
Inherited properties
Local convexity
If
is
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
then so is the
-topology on
and if
is a family of continuous seminorms generating this topology on
then the
-topology is induced by the following family of seminorms:
as
varies over
and
varies over
.
Hausdorffness
If
is
Hausdorff and
then the
-topology on
is Hausdorff.
Suppose that
is a topological space.
If
is
Hausdorff and
is the vector subspace of
consisting of all continuous maps that are bounded on every
and if
is dense in
then the
-topology on
is Hausdorff.
Boundedness
A subset
of
is
bounded in the
-topology if and only if for every
is bounded in
Examples of đť’˘-topologies
Pointwise convergence
If we let
be the set of all finite subsets of
then the
-topology on
is called the topology of pointwise convergence.
The topology of pointwise convergence on
is identical to the subspace topology that
inherits from
when
is endowed with the usual
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
.
If
is a non-trivial
completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
Hausdorff topological space and
is the space of all real (or complex) valued continuous functions on
the topology of pointwise convergence on
is
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 ...
if and only if
is countable.
đť’˘-topologies on spaces of continuous linear maps
Throughout this section we will assume that
and
are
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.
will be a non-empty collection of subsets of
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
by inclusion.
will denote the vector space of all continuous linear maps from
into
If
is given the
-topology inherited from
then this space with this topology is denoted by
.
The
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of a topological vector space
over the field
(which we will assume to be
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
) is the vector space
and is denoted by
.
The
-topology on
is compatible with the vector space structure of
if and only if for all
and all
the set
is bounded in
which we will assume to be the case for the rest of the article.
Note in particular that this is the case if
consists of
(von-Neumann) bounded subsets of
Assumptions on đť’˘
Assumptions that guarantee a vector topology
* (
is directed):
will be a non-empty collection of subsets of
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
by (subset) inclusion. That is, for any
there exists
such that
.
The above assumption guarantees that the collection of sets
forms a
filter base.
The next assumption will guarantee that the sets
are
balanced.
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
* (
are balanced):
is a neighborhoods basis of the origin in
that consists entirely of
balanced sets.
The following assumption is very commonly made because it will guarantee that each set
is absorbing in
* (
are bounded):
is assumed to consist entirely of bounded subsets of
The next theorem gives ways in which
can be modified without changing the resulting
-topology on
Common assumptions
Some authors (e.g. Narici) require that
satisfy the following condition, which implies, in particular, that
is
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
by subset inclusion:
:
is assumed to be closed with respect to the formation of subsets of finite unions of sets in
(i.e. every subset of every finite union of sets in
belongs to
).
Some authors (e.g. Trèves) require that
be directed under subset inclusion and that it satisfy the following condition:
:If
and
is a scalar then there exists a
such that
If
is a
bornology
In mathematics, especially functional analysis, a bornology on a set ''X'' is a collection of subsets of ''X'' satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is ...
on
which is often the case, then these axioms are satisfied.
If
is a
saturated family of
bounded subsets of
then these axioms are also satisfied.
Properties
Hausdorffness
A subset of a TVS
whose
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
is a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of
is said to be a
total subset In mathematics, more specifically in functional analysis, a subset T of a topological vector space X is said to be a total subset of X if the linear span of T is a dense subset of X.
This condition arises frequently in many theorems of functional ...
of
If
is a family of subsets of a TVS
then
is said to be
total in if the
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of
is dense in
If
is the vector subspace of
consisting of all continuous linear maps that are bounded on every
then the
-topology on
is Hausdorff if
is Hausdorff and
is total in
Completeness
For the following theorems, suppose that
is a topological vector space and
is a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
Hausdorff spaces and
is a collection of bounded subsets of
that covers
is directed by subset inclusion, and satisfies the following condition: if
and
is a scalar then there exists a
such that
- is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
if
- If is a Mackey space then is complete if and only if both and are complete.
- If is barrelled then is Hausdorff and
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
.
- Let and be TVSs with
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
and assume that (1) is barreled, or else (2) is a Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
and and are locally convex. If covers then every closed equicontinuous subset of is complete in and is quasi-complete.
- Let be a
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
, a locally convex space, and a family of bounded subsets of such that the range of every null sequence in is contained in some If is quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
(respectively, complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
) then so is .
Boundedness
Let
and
be topological vector spaces and
be a subset of
Then the following are equivalent:
- is bounded in ;
- For every is bounded in ;
- For every neighborhood of the origin in the set absorbs every
If
is a collection of bounded subsets of
whose union is
total in
then every
equicontinuous subset of
is bounded in the
-topology.
Furthermore, if
and
are locally convex Hausdorff spaces then
- if is bounded in (that is, pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of
- if is
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
(meaning that closed and bounded subsets are complete), then the bounded subsets of are identical for all -topologies where is any family of bounded subsets of covering
Examples
The topology of pointwise convergence
By letting
be the set of all finite subsets of
will have the weak topology on
or the topology of pointwise convergence or the topology of simple convergence and
with this topology is denoted by
.
Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity; for this reason, this article will avoid referring to this topology by this name.
A subset of
is called simply bounded or weakly bounded if it is bounded in
.
The weak-topology on
has the following properties:
- If is separable (that is, it has a countable dense subset) and if is a metrizable topological vector space then every equicontinuous subset of is metrizable; if in addition is separable then so is
* So in particular, on every equicontinuous subset of the topology of pointwise convergence is metrizable.
- Let denote the space of all functions from into If is given the topology of pointwise convergence then space of all linear maps (continuous or not) into is closed in .
* In addition, is dense in the space of all linear maps (continuous or not) into
- Suppose and are locally convex. Any simply bounded subset of is bounded when has the topology of uniform convergence on convex, balanced, bounded, complete subsets of If in addition is
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
then the families of bounded subsets of are identical for all -topologies on such that is a family of bounded sets covering
Equicontinuous subsets
- The weak-closure of an equicontinuous subset of is equicontinuous.
- If is locally convex, then the convex balanced hull of an equicontinuous subset of is equicontinuous.
- Let and be TVSs and assume that (1) is barreled, or else (2) is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
and and are locally convex. Then every simply bounded subset of is equicontinuous.
- On an equicontinuous subset of the following topologies are identical: (1) topology of pointwise convergence on a total subset of ; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.
Compact convergence
By letting
be the set of all compact subsets of
will have the topology of compact convergence or the topology of uniform convergence on compact sets and
with this topology is denoted by
.
The topology of compact convergence on
has the following properties:
- If is a
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
or a LF-space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces.
This means that ''X'' is a direct li ...
and if is a complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
locally convex Hausdorff space then is complete.
- On equicontinuous subsets of the following topologies coincide:
* The topology of pointwise convergence on a dense subset of
* The topology of pointwise convergence on
* The topology of compact convergence.
* The topology of precompact convergence.
- If is a
Montel space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a Barrelled space, barrelled topo ...
and is a topological vector space, then and have identical topologies.
Topology of bounded convergence
By letting
be the set of all bounded subsets of
will have the topology of bounded convergence on
or the topology of uniform convergence on bounded sets and
with this topology is denoted by
.
The topology of bounded convergence on
has the following properties:
- If is a
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
and if is a complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
locally convex Hausdorff space then is complete.
- If and are both normed spaces then the topology on induced by the usual operator norm is identical to the topology on .
* In particular, if is a normed space then the usual norm topology on the continuous dual space is identical to the topology of bounded convergence on .
- Every equicontinuous subset of is bounded in .
Polar topologies
Throughout, we assume that
is a TVS.
đť’˘-topologies versus polar topologies
If
is a TVS whose
bounded subsets are exactly the same as its bounded subsets (e.g. if
is a Hausdorff locally convex space), then a
-topology on
(as defined in this article) is a
polar topology
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairin ...
and conversely, every polar topology if a
-topology.
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if
is a TVS whose bounded subsets are exactly the same as its bounded subsets, then the notion of "bounded in
" is stronger than the notion of "
-bounded in
" (i.e. bounded in
implies
-bounded in
) so that a
-topology on
(as defined in this article) is necessarily a polar topology.
One important difference is that polar topologies are always locally convex while
-topologies need not be.
Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article:
polar topology
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairin ...
.
We list here some of the most common polar topologies.
List of polar topologies
Suppose that
is a TVS whose bounded subsets are the same as its weakly bounded subsets.
Notation: If
denotes a polar topology on
then
endowed with this topology will be denoted by
or simply
(e.g. for
we would have
so that
and
all denote
with endowed with
).
đť’˘-â„‹ topologies on spaces of bilinear maps
We will let
denote the space of separately continuous bilinear maps and
denote the space of continuous bilinear maps, where
and
are topological vector space over the same field (either the real or complex numbers).
In an analogous way to how we placed a topology on
we can place a topology on
and
.
Let
(respectively,
) be a family of subsets of
(respectively,
) containing at least one non-empty set.
Let
denote the collection of all sets
where
We can place on
the
-topology, and consequently on any of its subsets, in particular on
and on
.
This topology is known as the
-topology or as the topology of uniform convergence on the products
of
.
However, as before, this topology is not necessarily compatible with the vector space structure of
or of
without the additional requirement that for all bilinear maps,
in this space (that is, in
or in
) and for all
and
the set
is bounded in
If both
and
consist of bounded sets then this requirement is automatically satisfied if we are topologizing
but this may not be the case if we are trying to topologize
.
The
-topology on
will be compatible with the vector space structure of
if both
and
consists of bounded sets and any of the following conditions hold:
*
and
are barrelled spaces and
is locally convex.
*
is a
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 ...
,
is metrizable, and
is Hausdorff, in which case
*
and
are the strong duals of reflexive Fréchet spaces.
*
is normed and
and
the strong duals of reflexive Fréchet spaces.
The ε-topology
Suppose that
and
are locally convex spaces and let
and
be the collections of
equicontinuous subsets of
and
, respectively.
Then the
-topology on
will be a topological vector space topology.
This topology is called the ε-topology and
with this topology it is denoted by
or simply by
Part of the importance of this vector space and this topology is that it contains many subspace, such as
which we denote by
When this subspace is given the subspace topology of
it is denoted by
In the instance where
is the field of these vector spaces,
is 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
In fact, if
and
are locally convex Hausdorff spaces then
is vector space-isomorphic to
which is in turn is equal to
These spaces have the following properties:
* If
and
are locally convex Hausdorff spaces then
is complete if and only if both
and
are complete.
* If
and
are both normed (respectively, both Banach) then so is
See also
*
*
*
*
*
*
*
*
*
*
*
*
*
References
Bibliography
*
*
*
*
*
*
*
{{Topological vector spaces
Functional analysis
Topological vector spaces
Topology of function spaces