In
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 space#Definition, inner product, Norm (mathematics)#Defini ...
and related areas of
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 ...
, a metrizable (resp. pseudometrizable)
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) is a TVS whose topology is induced by a metric (resp.
pseudometric). An LM-space is an
inductive limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of a sequence of
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 ve ...
metrizable TVS.
Pseudometrics and metrics
A
pseudometric on a set
is a map
satisfying the following properties:
- ;
- Symmetry: ;
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
:
A pseudometric is called a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
if it satisfies:
Identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
: for all if then
Ultrapseudometric
A pseudometric
on
is called a
ultrapseudometric or a strong pseudometric if it satisfies:
- Strong/Ultrametric triangle inequality:
Pseudometric space
A
pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...
is a pair
consisting of a set
and a pseudometric
on
such that
's topology is identical to the topology on
induced by
We call a pseudometric space
a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
(resp. ultrapseudometric space) when
is a metric (resp. ultrapseudometric).
Topology induced by a pseudometric
If
is a pseudometric on a set
then collection of open balls:
as
ranges over
and
ranges over the positive real numbers,
forms a basis for a topology on
that is called the
-topology or the pseudometric topology on
induced by
:: If
is a pseudometric space and
is treated as a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, then unless indicated otherwise, it should be assumed that
is endowed with the topology induced by
Pseudometrizable space
A topological space
is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric)
on
such that
is equal to the topology induced by
Pseudometrics and values on topological groups
An additive
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
A topology
on a real or complex vector space
is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes
into 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 ...
).
Every
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)
is an additive commutative topological group but not all group topologies on
are vector topologies.
This is because despite it making addition and negation continuous, a group topology on a vector space
may fail to make scalar multiplication continuous.
For instance, the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
Translation invariant pseudometrics
If
is an additive group then we say that a pseudometric
on
is translation invariant or just invariant if it satisfies any of the following equivalent conditions:
Translation invariance
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
: ;
Value/G-seminorm
If
is a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
the a value or G-seminorm on
(the ''G'' stands for Group) is a real-valued map
with the following properties:
- Non-negative:
- Subadditive: ;
- Symmetric:
where we call a G-seminorm a G-norm if it satisfies the additional condition:
- Total/Positive definite: If then
Properties of values
If
is a value on a vector space
then:
- and for all and positive integers
- The set is an additive subgroup of
Equivalence on topological groups
Pseudometrizable topological groups
An invariant pseudometric that doesn't induce a vector topology
Let
be a non-trivial (i.e.
) real or complex vector space and let
be the translation-invariant trivial metric on
defined by
and
such that
The topology
that
induces on
is the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, which makes
into a commutative topological group under addition but does form a vector topology on
because
is
disconnected but every vector topology is connected.
What fails is that scalar multiplication isn't continuous on
This example shows that a translation-invariant (pseudo)metric is enough to guarantee a vector topology, which leads us to define paranorms and ''F''-seminorms.
Additive sequences
A collection
of subsets of a vector space is called additive if for every
there exists some
such that
All of the above conditions are consequently a necessary for a topology to form a vector topology.
Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
functions.
These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s.
Assume that
always denotes a finite sequence of non-negative integers and use the notation:
For any integers
and
From this it follows that if
consists of distinct positive integers then
It will now be shown by induction on
that if
consists of non-negative integers such that
for some integer
then
This is clearly true for
and
so assume that
which implies that all
are positive.
If all
are distinct then this step is done, and otherwise pick distinct indices
such that
and construct
from
by replacing each
with
and deleting the
element of
(all other elements of
are transferred to
unchanged).
Observe that
and
(because
) so by appealing to the inductive hypothesis we conclude that
as desired.
It is clear that
and that
so to prove that
is subadditive, it suffices to prove that
when
are such that
which implies that
This is an exercise.
If all
are symmetric then
if and only if
from which it follows that
and
If all
are balanced then the inequality
for all unit scalars
such that
is proved similarly.
Because
is a nonnegative subadditive function satisfying
as described in the article on
sublinear functional In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
s,
is uniformly continuous on
if and only if
is continuous at the origin.
If all
are neighborhoods of the origin then for any real
pick an integer
such that
so that
implies
If the set of all
form basis of balanced neighborhoods of the origin then it may be shown that for any
there exists some
such that
implies
Paranorms
If
is a vector space over the real or complex numbers then a paranorm on
is a G-seminorm (defined above)
on
that satisfies any of the following additional conditions, each of which begins with "for all sequences
in
and all convergent sequences of scalars
":
- Continuity of multiplication: if is a scalar and are such that and then
- Both of the conditions:
* if and if is such that then ;
* if then for every scalar
- Both of the conditions:
* if and for some scalar then ;
* if then
- Separate continuity:
* if for some scalar then for every ;
* if is a scalar, and then .
A paranorm is called total if in addition it satisfies:
- Total/Positive definite: implies
Properties of paranorms
If
is a paranorm on a vector space
then the map
defined by
is a translation-invariant pseudometric on
that defines a on
If
is a paranorm on a vector space
then:
- the set is a vector subspace of
- with
- If a paranorm satisfies and scalars then is absolutely homogeneity (i.e. equality holds) and thus is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
.
Examples of paranorms
- If is a translation-invariant pseudometric on a vector space that induces a vector topology on (i.e. is a TVS) then the map defines a continuous paranorm on ; moreover, the topology that this paranorm defines in is
- If is a paranorm on then so is the map
- Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
- Every
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is a paranorm.
- The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).
- The sum of two paranorms is a paranorm.
- If and are paranorms on then so is Moreover, and This makes the set of paranorms on into a conditionally complete lattice.
- Each of the following real-valued maps are paranorms on :
*
*
- The real-valued maps and are a paranorms on
- If is a
Hamel basis
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
on a vector space then the real-valued map that sends (where all but finitely many of the scalars are 0) to is a paranorm on which satisfies for all and scalars
- The function is a paranorm on that is balanced but nevertheless equivalent to the usual norm on Note that the function is subadditive.
- Let be a complex vector space and let denote considered as a vector space over Any paranorm on is also a paranorm on
-
''F''-seminorms
If
is a vector space over the real or complex numbers then an ''F''-seminorm on
(the
stands for
Fréchet) is a real-valued map
with the following properties:
- Non-negative:
- Subadditive: ;
- Balanced: for all and all scalars satisfying ;
* This condition guarantees that each set of the form or for some is
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 ...
.
- for every as
* The sequence can be replaced by any positive sequence converging to 0.
An ''F''-seminorm is called an ''F''-norm if in addition it satisfies:
- Total/Positive definite: implies
An ''F''-seminorm is called monotone if it satisfies:
- Monotone: for all non-zero and all real and such that
''F''-seminormed spaces
An ''F''-seminormed space (resp. ''F''-normed space) is a pair
consisting of a vector space
and an ''F''-seminorm (resp. ''F''-norm)
on
If
and
are ''F''-seminormed spaces then a map
is called an isometric embedding if
Every isometric embedding of one ''F''-seminormed space into another is a
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 ...
, but the converse is not true in general.
Examples of ''F''-seminorms
- Every positive scalar multiple of an ''F''-seminorm (resp. ''F''-norm, seminorm) is again an ''F''-seminorm (resp. ''F''-norm, seminorm).
- The sum of finitely many ''F''-seminorms (resp. ''F''-norms) is an ''F''-seminorm (resp. ''F''-norm).
- If and are ''F''-seminorms on then so is their pointwise supremum The same is true of the supremum of any non-empty finite family of ''F''-seminorms on
- The restriction of an ''F''-seminorm (resp. ''F''-norm) to a vector subspace is an ''F''-seminorm (resp. ''F''-norm).
- A non-negative real-valued function on is a seminorm if and only if it is a
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 polytope ...
''F''-seminorm, or equivalently, if and only if it is a convex balanced ''G''-seminorm. In particular, every seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is an ''F''-seminorm.
- For any the map on defined by
is an ''F''-norm that is not a norm.
- If is a linear map and if is an ''F''-seminorm on then is an ''F''-seminorm on
- Let be a complex vector space and let denote considered as a vector space over Any ''F''-seminorm on is also an ''F''-seminorm on
Properties of ''F''-seminorms
Every ''F''-seminorm is a paranorm and every paranorm is equivalent to some ''F''-seminorm.
Every ''F''-seminorm on a vector space
is a value on
In particular,
and
for all
Topology induced by a single ''F''-seminorm
Topology induced by a family of ''F''-seminorms
Suppose that
is a non-empty collection of ''F''-seminorms on a vector space
and for any finite subset
and any
let
The set
forms a filter base on
that also forms a neighborhood basis at the origin for a vector topology on
denoted by
Each
is a
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 ...
and
absorbing subset of
These sets satisfy
- is the coarsest vector topology on making each continuous.
- is Hausdorff if and only if for every non-zero there exists some such that
- If is the set of all continuous ''F''-seminorms on then
- If is the set of all pointwise suprema of non-empty finite subsets of of then is a directed family of ''F''-seminorms and
Fréchet combination
Suppose that
is a family of non-negative subadditive functions on a vector space
The Fréchet combination of
is defined to be the real-valued map
As an ''F''-seminorm
Assume that
is an increasing sequence of seminorms on
and let
be the Fréchet combination of
Then
is an ''F''-seminorm on
that induces the same locally convex topology as the family
of seminorms.
Since
is increasing, a basis of open neighborhoods of the origin consists of all sets of the form
as
ranges over all positive integers and
ranges over all positive real numbers.
The
translation invariant
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equat ...
pseudometric on
induced by this ''F''-seminorm
is
This metric was discovered by
Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.
As a paranorm
If each
is a paranorm then so is
and moreover,
induces the same topology on
as the family
of paranorms.
This is also true of the following paranorms on
:
Generalization
The Fréchet combination can be generalized by use of a bounded remetrization function.
A bounded remetrization function is a continuous non-negative non-decreasing map