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, norm, topology, etc.) and the linear functions defined o ...
and related areas of
mathematics, a complete topological vector space 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 ...
(TVS) with the property that whenever points get progressively closer to each other, then there exists some point
towards which they all get closer.
The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point
towards which they all get closer" means that this Cauchy
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 ...
or filter
converges to
The notion of completeness for TVSs uses the theory of
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s as a framework to generalize the notion of
completeness for metric spaces.
But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for TVSs, including those that are not
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 ...
or
Hausdorff.
Completeness is an extremely important property for a topological vector space to possess.
The notions of completeness for
normed spaces and
metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
s, which are commonly defined in terms of
completeness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness – a notion that is independent of any particular norm or metric.
A
metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
with a
translation invariant metric[A metric on a vector space is said to be translation invariant if for all vectors A metric that is induced by a norm is always translation invariant.] is complete as a TVS if and only if
is a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, which by definition means that every
-
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
converges to some point in
Prominent examples of complete TVSs that are also
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 ...
include all
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 ...
s and consequently also all
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 ...
s,
Banach spaces, and
Hilbert spaces.
Prominent examples of complete TVS that are (typically) metrizable include strict
LF-spaces such as the
space of test functions with it canonical LF-topology, the
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of any non-normable
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 ...
, as well as many other
polar topologies on
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 ...
or other
topologies on spaces of linear maps.
Explicitly, a
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 ...
(TVS) is complete if every
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 ...
, or equivalently, every
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
, that is
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
with respect to the space's necessarily converges to some point. Said differently, a TVS is complete if its canonical uniformity is a
complete uniformity.
The canonical uniformity on a TVS
is the unique
[Completeness of normed spaces and ]metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
s are defined in terms of norms and metrics
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 ...
. In general, many different norms (for example, equivalent norms) and metrics may be used to determine completeness of such space. This stands in contrast to the uniqueness of this translation-invariant canonical uniformity. translation-invariant
uniformity that induces on
the topology
This notion of "TVS-completeness" depends on vector subtraction and the topology of the TVS; consequently, it can be applied to all TVSs, including those whose topologies can not be defined in terms
metrics
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 ...
or
pseudometrics.
A
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
TVS is complete if and only if every Cauchy sequence (or equivalently, every
elementary
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, a ...
Cauchy filter) converges to some point.
Every topological vector space
even if it is not
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 ...
or not
Hausdorff, has a , which by definition is a complete TVS
into which
can be
TVS-embedded as a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
. Moreover, every Hausdorff TVS has a completion, which is necessarily unique
up to TVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that are TVS-isomorphic to one another.
Definitions
This section summarizes the definition of a complete
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) in terms of both
nets and
prefilters.
Information about convergence of nets and filters, such as definitions and properties, can be found in the article about
filters in topology
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some give ...
.
Every topological vector space (TVS) is a commutative
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 st ...
with identity under addition and the canonical uniformity of a TVS is defined in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.
Canonical uniformity
The of
is the set
and for any
the / is the set
where if
then
contains the diagonal
If
is a
symmetric set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to ...
(that is, if
), then
is , which by definition means that
holds where
and in addition, this symmetric set's with itself is:
If
is any neighborhood basis at the origin in
then the
family of subsets of
is a
prefilter on
If
is the
neighborhood filter at the origin in
then
forms a
base of entourages for a
uniform structure
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
on
that is considered
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
.
Explicitly, by definition,
is the
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on
generated by the above prefilter:
where
denotes the of
in
The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. If
is any neighborhood basis at the origin in
then the filter on
generated by the prefilter
is equal to the canonical uniformity
induced by
Cauchy net
The general theory of
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on
these definitions reduce down to those given below.
Suppose
is a net in
and
is a net in
The product
becomes a
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 ...
by declaring
if and only if
and
Then
denotes the (Cartesian) , where in particular
If
then the image of this net under the vector addition map
denotes the of these two nets:
and similarly their is defined to be the image of the product net under the vector subtraction map
:
In particular, the notation
denotes the
-indexed net
and not the
-indexed net
since using the latter as the definition would make the notation useless.
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 a TVS
is called a Cauchy net if
Explicitly, this means that for every neighborhood
of
in
there exists some index
such that
for all indices
that satisfy
and
It suffices to check any of these defining conditions for any given
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 ...
of
in
A
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
is a sequence that is also a Cauchy net.
If
then
in
and so the continuity of the vector subtraction map
which is defined by
guarantees that
in
where
and
This proves that every convergent net is a Cauchy net.
By definition, a space is called if the converse is also always true.
That is,
is complete if and only if the following holds:
:whenever
is a net in
then
converges (to some point) in
if and only if
in
A similar characterization of completeness holds if filters and prefilters are used instead of nets.
A series
is called a (respectively, a ) if the sequence of
partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s
is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
(respectively, a
convergent sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...
). Every convergent series is necessarily a Cauchy series. In a complete TVS, every Cauchy series is necessarily a convergent series.
Cauchy filter and Cauchy prefilter
A
prefilter on an
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 ...
is called a Cauchy prefilter if it satisfies any of the following equivalent conditions:
- in
* The family is a prefilter.
* Explicitly, means that for every neighborhood of the origin in there exist such that
- in
* The family is a prefilter equivalent to (''equivalence'' means these prefilters generate the same filter on ).
* Explicitly, means that for every neighborhood of the origin in there exists some such that
- For every neighborhood of the origin in contains some -small set (that is, there exists some such that ).
* A subset is called -small or if
- For every neighborhood of the origin in there exists some and some such that
* This statement remains true if "" is replaced with ""
- Every neighborhood of the origin in contains some subset of the form where and
It suffices to check any of the above conditions for any given
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 ...
of
in
A
Cauchy filter is a Cauchy prefilter that is also a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on
If
is a prefilter on a topological vector space
and if
then
in
if and only if
and
is Cauchy.
Complete subset
For any
a prefilter
is necessarily a subset of
; that is,
A subset
of a TVS
is called a if it satisfies any of the following equivalent conditions:
- Every Cauchy prefilter on converges to at least one point of
* If is Hausdorff then every prefilter on will converge to at most one point of But if is not Hausdorff then a prefilter may converge to multiple points in The same is true for nets.
- Every Cauchy net in converges to at least one point of
- is a complete uniform space (under the point-set topology definition of " complete uniform space") when is endowed with the uniformity induced on it by the canonical uniformity of
The subset
is called a if every Cauchy sequence in
(or equivalently, every elementary Cauchy filter/prefilter on
) converges to at least one point of
Importantly, : If
is not Hausdorff and if every Cauchy prefilter on
converges to some point of
then
will be complete even if some or all Cauchy prefilters on
converge to points(s) in
In short, there is no requirement that these Cauchy prefilters on
converge to points in
The same can be said of the convergence of Cauchy nets in
As a consequence, if a TVS
is Hausdorff then every subset of the closure of
in
is complete because it is compact and every compact set is necessarily complete. In particular, if
is a proper subset, such as
for example, then
would be complete even though Cauchy net in
(and also every Cauchy prefilter on
) converges to point in
including those points in
that do not belong to
This example also shows that complete subsets (and indeed, even compact subsets) of a non-Hausdorff TVS may fail to be closed. For example, if
then
if and only if
is closed in
Complete topological vector space
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 ...
is called a if any of the following equivalent conditions are satisfied:
- is a complete uniform space when it is endowed with its canonical uniformity.
* In the general theory of
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s, a uniform space is called a complete uniform space if each Cauchy filter on converges to some point of in the topology induced by the uniformity. When is a TVS, the topology induced by the canonical uniformity is equal to 's given topology (so convergence in this induced topology is just the usual convergence in ).
- is a complete subset of itself.
- There exists a neighborhood of the origin in that is also a complete subset of
* This implies that every locally compact TVS is complete (even if the TVS is not Hausdorff).
- Every Cauchy prefilter on converges in to at least one point of
* If is Hausdorff then every prefilter on will converge to at most one point of But if is not Hausdorff then a prefilter may converge to multiple points in The same is true for nets.
- Every Cauchy
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on converges in to at least one point of
- Every Cauchy net in converges in to at least one point of
where if in addition
is
pseudometrizable or metrizable (for example, a
normed space) then this list can be extended to include:
- is sequentially complete.
A topological vector space
is if any of the following equivalent conditions are satisfied:
- is a sequentially complete subset of itself.
- Every Cauchy sequence in converges in to at least one point of
- Every elementary Cauchy prefilter on converges in to at least one point of
- Every elementary Cauchy filter on converges in to at least one point of
Uniqueness of the canonical uniformity
The existence of the canonical uniformity was demonstrated above by defining it. The theorem below establishes that the canonical uniformity of any TVS
is the only uniformity on
that is both (1) translation invariant, and (2) generates on
the topology
This section is dedicated to explaining the precise meanings of the terms involved in this uniqueness statement.
Uniform spaces and translation-invariant uniformities
For any subsets
let
and let
A non-empty family
is called a or a if
is a
prefilter on
satisfying all of the following conditions:
- Every set in contains the diagonal of as a subset; that is, for every Said differently, the prefilter is on
- For every there exists some such that
- For every there exists some such that
A or on
is a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on
that is generated by some base of entourages
in which case we say that
is a base of entourages
For a commutative additive group
a is a fundamental system of entourages
such that for every
if and only if
for all
A uniformity
is called a if it has a base of entourages that is translation-invariant.
The canonical uniformity on any TVS is translation-invariant.
The binary operator
satisfies all of the following:
- If and then
- Associativity:
- Identity:
- Zero:
Symmetric entourages
Call a subset
symmetric if
which is equivalent to
This equivalence follows from the identity
and the fact that if
then
if and only if
For example, the set
is always symmetric for every
And because
if
and
are symmetric then so is
Topology generated by a uniformity
Relatives
Let
be arbitrary and let
be the canonical projections onto the first and second coordinates, respectively.
For any
define
where
(respectively,
) is called the set of left (respectively, right)
-relatives of (points in)
Denote the special case where
is a singleton set for some
by:
If
then
Moreover,
right distributes over both unions and intersections, meaning that if
then
and
Neighborhoods and open sets
Two points
and
are
-close if
and a subset
is called
-small if
Let
be a base of entourages on
The at a point
and, respectively, on a subset
are the
families of sets:
and the filters on
that each generates is known as the of
(respectively, of
).
Assign to every
the neighborhood prefilter
and use the
neighborhood definition of "open set" to obtain a
topology
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 ...
on
called the topology induced by
or the .
Explicitly, a subset
is open in this topology if and only if for every
there exists some
such that
that is,
is open if and only if for every
there exists some
such that
The closure of a subset
in this topology is:
Cauchy prefilters and complete uniformities
A prefilter
on a uniform space
with uniformity
is called a Cauchy prefilter if for every entourage
there exists some
such that
A uniform space
is called a (respectively, a ) if every Cauchy prefilter (respectively, every elementary Cauchy prefilter) on
converges to at least one point of
when
is endowed with the topology induced by
Case of a topological vector space
If
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 ...
then for any
and
and the topology induced on
by the canonical uniformity is the same as the topology that
started with (that is, it is
).
Uniform continuity
Let
and
be TVSs,
and
be a map. Then
is if for every neighborhood
of the origin in
there exists a neighborhood
of the origin in
such that for all
if
then
Suppose that
is uniformly continuous. If
is a Cauchy net in
then
is a Cauchy net in
If
is a Cauchy prefilter in
(meaning that
is a family of subsets of
that is Cauchy in
) then
is a Cauchy prefilter in
However, if
is a Cauchy filter on
then although
will be a Cauchy filter, it will be a Cauchy filter in
if and only if
is surjective.
TVS completeness vs completeness of (pseudo)metrics
Preliminaries: Complete pseudometric spaces
We review the basic notions related to the general theory of complete pseudometric spaces.
Recall that every
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 mathem ...
is a
pseudometric and that a pseudometric
is a metric if and only if
implies
Thus every
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 set ...
is 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 ...
and a pseudometric space
is a metric space if and only if
is a metric.
If
is a subset of 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 ...
then the
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
of
is defined to be
A prefilter
on a pseudometric space
is called a
-Cauchy prefilter or simply a Cauchy prefilter if for each
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) ...
there is some
such that the diameter of
is less than
Suppose
is a pseudometric space. 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
is called a
-Cauchy net or simply a Cauchy net if
is a Cauchy prefilter, which happens if and only if
:for every
there is some
such that if
with
and
then
or equivalently, if and only if
in
This is analogous to the following characterization of the converge of
to a point: if
then
in
if and only if
in
A
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
is a sequence that is also a Cauchy net.
[Every sequence is also a net.]
Every pseudometric
on a set
induces the usual canonical topology on
which we'll denote by
; it also induces a canonical
uniformity on
which we'll denote by
The topology on
induced by the uniformity
is equal to
A net
in
is Cauchy with respect to
if and only if it is Cauchy with respect to the uniformity
The pseudometric space
is a
complete (resp. a sequentially complete) pseudometric space if and only if
is a
complete (resp. a sequentially complete) uniform space.
Moreover, the pseudometric space
(resp. the uniform space
) is complete if and only if it is sequentially complete.
A pseudometric space
(for example, 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 set ...
) is called complete and
is called a complete pseudometric if any of the following equivalent conditions hold:
- Every Cauchy prefilter on converges to at least one point of
- The above statement but with the word "prefilter" replaced by "filter."
- Every Cauchy net in converges to at least one point of
* If is a metric on then any limit point is necessarily unique and the same is true for limits of Cauchy prefilters on
- Every Cauchy sequence in converges to at least one point of
* Thus to prove that is complete, it suffices to only consider Cauchy sequences in (and it is not necessary to consider the more general Cauchy nets).
- The canonical uniformity on induced by the pseudometric is a complete uniformity.
And if addition
is a metric then we may add to this list:
- Every decreasing sequence of closed balls whose diameters shrink to has non-empty intersection.
Complete pseudometrics and complete TVSs
Every
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 ...
, and thus also every
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 ...
,
Banach space, and
Hilbert space is a complete TVS. Note that every ''F''-space 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 ...
but there are normed spaces that are Baire but not Banach.
A pseudometric
on a vector space
is said to be a if
for all vectors
Suppose
is
pseudometrizable TVS (for example, a metrizable TVS) and that
is pseudometric on
such that the topology on
induced by
is equal to
If
is translation-invariant, then
is a complete TVS if and only if
is a complete pseudometric space.
If
is translation-invariant, then may be possible for
to be a complete TVS but
to be a complete pseudometric space (see this footnote
[The normed space is a Banach space where the absolute value is a norm that induces the usual Euclidean topology on Define a metric on by for all where one may show that induces the usual Euclidean topology on However, is not a complete metric since the sequence defined by is a -Cauchy sequence that does not converge in to any point of Note also that this -Cauchy sequence is not a Cauchy sequence in (that is, it is not a Cauchy sequence with respect to the norm ).] for an example).
Complete norms and equivalent norms
Two norms on a vector space are called
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
if and only if they induce the same topology.
If
and
are two equivalent norms on a vector space
then the
normed space is a
Banach space if and only if
is a Banach space.
See this footnote for an example of a continuous norm on a Banach space that is not is equivalent to that Banach space's given norm.
[Let denotes the Banach space of continuous functions with the supremum norm, let where is given the topology induced by and denote the restriction of the L1-norm to by Then one may show that so that the norm is a continuous function. However, is equivalent to the norm and so in particular, is a Banach space.]
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.
[see Corollary1.4.18, p.32 in .] Every Banach space is a complete TVS. A normed space is a Banach space (that is, its canonical norm-induced metric is complete) if and only if it is complete as a topological vector space.
Completions
A completion of a TVS
is a complete TVS that contains a dense vector subspace that is TVS-isomorphic to
In other words, it is a complete TVS
into which
can be
TVS-embedded as a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
. Every TVS-embedding is a
uniform embedding.
Every topological vector space has a completion. Moreover, every Hausdorff TVS has a completion, which is necessarily unique
up to TVS-isomorphism. However, all TVSs, even those that are Hausdorff, (already) complete, and/or metrizable have infinitely many non-Hausdorff completions that are TVS-isomorphic to one another.
Examples of completions
For example, the vector space consisting of scalar-valued
simple functions
for which
(where this seminorm is defined in the usual way in terms of
Lebesgue integration) becomes a
seminormed space when endowed with this seminorm, which in turn makes it into both 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 ...
and a non-Hausdorff non-complete TVS; any completion of this space is a non-Hausdorff complete seminormed space that when
quotiented by the closure of its origin (so as to
obtain a Hausdorff TVS) results in (a space
linearly isometrically-isomorphic to) the usual complete Hausdorff
-space (endowed with the usual complete
norm).
As another example demonstrating the usefulness of completions, the completions of
topological tensor product In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of ...
s, such as
projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times ...
s or
injective tensor product 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 it ...
s, of the Banach space
with a complete Hausdorff locally convex TVS
results in a complete TVS that is TVS-isomorphic to a "generalized"
-space consisting
-valued functions on
(where this "generalized" TVS is defined analogously to original space
of scalar-valued functions on
). Similarly, the completion of the injective tensor product of the
space of scalar-valued -test functions with such a TVS
is TVS-isomorphic to the analogously defined TVS of
-valued test functions.
Non-uniqueness of all completions
As the example below shows, regardless of whether or not a space is Hausdorff or already complete, 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) has infinitely many non-isomorphic completions.
However, every Hausdorff TVS has a completion that is unique up to TVS-isomorphism. But nevertheless, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.
Example (Non-uniqueness of completions):
Let
denote any complete TVS and let
denote any TVS endowed with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, which recall makes
into a complete TVS.
Since both
and
are complete TVSs, so is their product
If
and
are non-empty open subsets of
and
respectively, then
and
which shows that
is a dense subspace of
Thus by definition of "completion,"
is a completion of
(it doesn't matter that
is already complete).
So by identifying
with
if
is a dense vector subspace of
then
has both
and
as completions.
Hausdorff completions
Every Hausdorff TVS has a completion that is unique up to TVS-isomorphism. But nevertheless, as shown above, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.
Existence of Hausdorff completions
A Cauchy filter
on a TVS
is called a if there does exist a Cauchy filter on
that is strictly coarser than
(that is, "strictly coarser than
" means contained as a proper subset of
).
If
is a Cauchy filter on
then the filter generated by the following prefilter:
is the unique minimal Cauchy filter on
that is contained as a subset of
In particular, for any
the neighborhood filter at
is a minimal Cauchy filter.
Let
be the set of all minimal Cauchy filters on
and let
be the map defined by sending
to the neighborhood filter of
in
Endow
with the following vector space structure:
Given
and a scalar
let
(resp.
) denote the unique minimal Cauchy filter contained in the filter generated by
(resp.
).
For every
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 ...
neighborhood
of the origin in
let
If
is Hausdorff then the collection of all sets
as
ranges over all balanced neighborhoods of the origin in
forms a vector topology on
making
into a complete Hausdorff TVS. Moreover, the map
is a TVS-embedding onto a dense vector subspace of
If
is a
metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
then a Hausdorff completion of
can be constructed using equivalence classes of Cauchy sequences instead of minimal Cauchy filters.
Non-Hausdorff completions
This subsection details how every non-Hausdorff TVS
can be TVS-embedded onto a dense vector subspace of a complete TVS. The proof that every Hausdorff TVS has a Hausdorff completion is widely available and so this fact will be used (without proof) to show that every non-Hausdorff TVS also has a completion. These details are sometimes useful for extending results from Hausdorff TVSs to non-Hausdorff TVSs.
Let
denote the closure of the origin in
where
is endowed with its subspace topology induced by
(so that
has the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
).
Since
has the trivial topology, it is easily shown that every vector subspace of
that is an algebraic complement of
in
is necessarily a
topological complement of
in
Let
denote any topological complement of
in
which is necessarily a Hausdorff TVS (since it is TVS-isomorphic to the quotient TVS
[This particular quotient map is in fact also a closed map.]).
Since
is the
topological direct sum of
and
(which means that
in the category of TVSs), the canonical map
is a TVS-isomorphism.
Let
denote the inverse of this canonical map. (As a side note, it follows that every open and every closed subset
of
satisfies
[Let be a neighborhood of the origin in Since is a neighborhood of in there exists an open (resp. closed) neighborhood of in such that is a neighborhood of the origin. Clearly, is open (resp. closed) if and only if is open (resp. closed). Let so that where is open (resp. closed) if and only if is open (resp. closed).])
The Hausdorff TVS
can be TVS-embedded, say via the map
onto a dense vector subspace of its completion
Since
and
are complete, so is their product
Let
denote the identity map and observe that the product map
is a TVS-embedding whose image is dense in
Define the map
[Explicitly, this map is defined as follows: for each let and so that Then holds for all and ]
which is a TVS-embedding of
onto a dense vector subspace of the complete TVS
Moreover, observe that the closure of the origin in
is equal to
and that
and
are topological complements in
To summarize, given any algebraic (and thus topological) complement
of
in
and given any completion
of the Hausdorff TVS
such that
then the natural inclusion
[where for all and ]
is a well-defined TVS-embedding of
onto a dense vector subspace of the complete TVS
where moreover,
Topology of a completion
Said differently, if
is a completion of a TVS
with
and if
is a
neighborhood base 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 ...
of the origin in
then the family of sets
is a neighborhood basis at the origin in
Grothendieck's Completeness Theorem
Let
denote the on the continuous dual space
which by definition consists of all
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 ...
weak-* closed and weak-*
bounded absolutely convex subsets of
(which are necessarily weak-* compact subsets of
). Assume that every
is endowed with the
weak-* topology.
A
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on
is said to to
if there exists some
containing
(that is,
) such that the trace of
on
which is the family
converges to in
(that is, if
in the given weak-* topology).
The filter
converges continuously to
if and only if
converges continuously to the origin, which happens if and only if for every
the filter
in the scalar field (which is
or
) where
denotes any neighborhood basis at the origin in
denotes the
duality pairing, and
denotes the filter generated by
A map
into a topological space (such as
or
) is said to be if whenever a filter
on
converges continuously to
then
Properties preserved by completions
If a TVS
has any of the following properties then so does its completion:
- Hausdorff
- Locally convex
- Pseudometrizable
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 ...
- Seminormable
- Normable
* Moreover, if is a normed space, then the completion can be chosen to be a Banach space such that the TVS-embedding of into is an isometry.
- Hausdorff pre-Hilbert. That is, a TVS induced by an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
.
- Nuclear
- Barrelled
- Mackey
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 ...
Completions of Hilbert spaces
Every inner product space
has a completion
that is a Hilbert space, where the inner product
is the unique continuous extension to
of the original inner product
The norm induced by
is also the unique continuous extension to
of the norm induced by
Other preserved properties
If
is a
Hausdorff TVS, then the continuous dual space of
is identical to the continuous dual space of the completion of
The completion of a locally convex
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 ...
is a
barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
. If
and
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 then the
projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times ...
, as well as its completion, of these spaces is a DF-space.
The completion of the
projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times ...
of two nuclear spaces is nuclear. The completion of a nuclear space is TVS-isomorphic with a projective limit of
Hilbert spaces.
If
(meaning that the addition map
is a TVS-isomorphism) has a Hausdorff completion
then
If in addition
is an
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
and
and
are
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
s of each other in
(that is,
), then
and
are orthogonal complements in the
Hilbert space
Properties of maps preserved by extensions to a completion
If
is a
nuclear linear operator between two locally convex spaces and if
be a completion of
then
has a unique continuous linear extension to a nuclear linear operator
Let
and
be two Hausdorff TVSs with
complete. Let
be a completion of
Let
denote the vector space of continuous linear operators and let
denote the map that sends every
to its unique continuous linear extension on
Then
is a (surjective) vector space isomorphism. Moreover,
maps families of
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 onto each other. Suppose that
is endowed with a
-topology and that
denotes the closures in
of sets in
Then the map
is also a TVS-isomorphism.
Examples and sufficient conditions for a complete TVS
- Any TVS endowed with the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
is complete and every one of its subsets is complete. Moreover, every TVS with the trivial topology is compact and hence locally compact. Thus a complete seminormable locally convex and locally compact TVS need not be finite-dimensional if it is not Hausdorff.
- An arbitrary product of complete (resp. sequentially complete, quasi-complete) TVSs has that same property. If all spaces are Hausdorff, then the converses are also true. A product of Hausdorff completions of a family of (Hausdorff) TVSs is a Hausdorff completion of their product TVS. More generally, an arbitrary product of complete subsets of a family of TVSs is a complete subset of the product TVS.
- The projective limit of a projective system of Hausdorff complete (resp. sequentially complete, quasi-complete) TVSs has that same property. A projective limit of Hausdorff completions of an inverse system of (Hausdorff) TVSs is a Hausdorff completion of their projective limit.
- If is a closed vector subspace of a complete pseudometrizable TVS then the quotient space is complete.
- Suppose is a vector subspace of a
metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
If the quotient space is complete then so is However, there exists a complete TVS having a closed vector subspace such that the quotient TVS is complete.
- Every
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 ...
, 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 ...
, Banach space, and Hilbert space is a complete TVS.
- Strict LF-spaces and strict
LB-space In mathematics, an ''LB''-space, also written (''LB'')-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Banach spaces.
This means that X is a direct limit of a direct syste ...
s are complete.
- Suppose that is a dense subset of a TVS If every Cauchy filter on converges to some point in then is complete.
- The
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
of smooth functions is complete.
- The spaces of distributions and test functions is complete.
- Suppose that and are locally convex TVSs and that the space of continuous linear maps is endowed with the topology of uniform convergence on bounded subsets of 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 complete then is a complete TVS. In particular, the strong dual of 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 ...
is complete. However, it need not be bornological.
- Every
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 ...
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 ...
is complete.
- Let and be Hausdorff TVS topologies on a vector space such that If there exists a prefilter such that is 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 and such that every is a complete subset of then is a complete TVS.
Properties
Complete TVSs
Every TVS has a
completion and every Hausdorff TVS has a Hausdorff completion.
Every complete TVS is
quasi-complete space
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
* Ev ...
and
sequentially complete.
However, the converses of the above implications are generally false.
There exists a
sequentially complete locally convex TVS that is not
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 ...
.
If a TVS has a complete neighborhood of the origin then it is complete.
Every complete
metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is a
barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
and 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 thus non-meager).
The dimension of a complete metrizable TVS is either finite or uncountable.
Cauchy nets and prefilters
Any
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 ...
of any point in a TVS is a Cauchy prefilter.
Every convergent net (respectively, prefilter) in a TVS is necessarily a Cauchy net (respectively, a Cauchy prefilter).
Any prefilter that is subordinate to (that is, finer than) a Cauchy prefilter is necessarily also a Cauchy prefilter and any prefilter finer than a Cauchy prefilter is also a Cauchy prefilter.
The filter associated with a sequence in a TVS is Cauchy if and only if the sequence is a Cauchy sequence.
Every convergent prefilter is a Cauchy prefilter.
If
is a TVS and if
is a cluster point of a Cauchy net (respectively, Cauchy prefilter), then that Cauchy net (respectively, that Cauchy prefilter) converges to
in
If a Cauchy filter in a TVS has an
accumulation point then it converges to
Uniformly continuous maps send Cauchy nets to Cauchy nets.
A Cauchy sequence in a Hausdorff TVS
A Cauchy sequence in a Hausdorff TVS
when considered as a set, is not necessarily
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
(that is, its closure in
is not necessarily compact
[If is a normable TVS such that for every Cauchy sequence the closure of in is compact (and thus ]sequentially compact
In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X.
Every metric space is naturally a topological space, and for metric spaces, the notio ...
) then this guarantees that there always exist some such that in Thus any normed space with this property is necessarily sequentially complete. Since not all normed spaces are complete, the closure of a Cauchy sequence is not necessarily compact.) although it is precompact (that is, its closure in the completion of
is compact).
Every Cauchy sequence is a
bounded subset but this is not necessarily true of Cauchy net. For example, let
have it usual order, let
denote any
preorder on the non-
indiscrete TVS
(that is,
does not have the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
; it is also assumed that
) and extend these two preorders to the union
by declaring that
holds for every
and
Let
be defined by
if
and
otherwise (that is, if
), which is a net in
since the preordered set
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 ...
(this preorder on
is also
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
(respectively, a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflex ...
) if this is true of
). This net
is a Cauchy net in
because it converges to the origin, but the set
is not a bounded subset of
(because
does not have the trivial topology).
Suppose that
is a family of TVSs and that
denotes the product of these TVSs. Suppose that for every index
is a prefilter on
Then the product of this family of prefilters is a Cauchy filter on
if and only if each
is a Cauchy filter on
Maps
If
is an injective
topological homomorphism In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs).
This concept is of considerable importance in functional ana ...
from a complete TVS into a Hausdorff TVS then the image of
(that is,
) is a closed subspace of
If
is a
topological homomorphism In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs).
This concept is of considerable importance in functional ana ...
from a complete
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 ...
TVS into a Hausdorff TVS then the range of
is a closed subspace of
If
is a
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
map between two Hausdorff TVSs then the image under
of a totally bounded subset of
is a totally bounded subset of
Uniformly continuous extensions
Suppose that
is a uniformly continuous map from a dense subset
of a TVS
into a complete Hausdorff TVS
Then
has a unique uniformly continuous extension to all of
If in addition
is a homomorphism then its unique uniformly continuous extension is also a homomorphism.
This remains true if "TVS" is replaced by "commutative topological group."
The map
is not required to be a linear map and that
is not required to be a vector subspace of
Uniformly continuous linear extensions
Suppose
be a continuous linear operator between two Hausdorff TVSs. If
is a dense vector subspace of
and if the restriction
to
is a
topological homomorphism In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs).
This concept is of considerable importance in functional ana ...
then
is also a topological homomorphism. So if
and
are Hausdorff completions of
and
respectively, and if
is a topological homomorphism, then
's unique continuous linear extension
is a topological homomorphism. (Note that it's possible for
to be surjective but for
to be injective.)
Suppose
and
are Hausdorff TVSs,
is a dense vector subspace of
and
is a dense vector subspaces of
If
are and
are topologically isomorphic additive subgroups via a topological homomorphism
then the same is true of
and
via the unique uniformly continuous extension of
(which is also a homeomorphism).
Subsets
Complete subsets
Every complete subset of a TVS is
sequentially complete.
A complete subset of a Hausdorff TVS
is a closed subset of
Every compact subset of a TVS is complete (even if the TVS is not Hausdorff or not complete).
Closed subsets of a complete TVS are complete; however, if a TVS
is not complete then
is a closed subset of
that is not complete.
The empty set is complete subset of every TVS.
If
is a complete subset of a TVS (the TVS is not necessarily Hausdorff or complete) then any subset of
that is closed in
is complete.
Topological complements
If
is a non-normable
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 ...
on which there exists a continuous norm then
contains a closed vector subspace that has no
topological complement.
If
is a complete TVS and
is a closed vector subspace of
such that
is not complete, then
does have a
topological complement in
Subsets of completions
Let
be a
separable locally convex
metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
and let
be its completion. If
is a bounded subset of
then there exists a bounded subset
of
such that
Relation to compact subsets
A subset of a TVS ( assumed to be Hausdorff or complete) is
compact if and only if it is complete and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...
.
[Suppose is compact in and let be a Cauchy filter on Let so that is a Cauchy filter of closed sets. Since has the finite intersection property, there exists some such that for all so { (that is, is an accumulation point of ). Since is Cauchy, in Thus is complete. That is also totally bounded follows immediately from the compactness of ]
Thus a closed and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...
subset of a complete TVS is compact.
In a Hausdorff locally convex TVS, the convex hull of a
precompact set is again precompact. Consequently, in a complete locally convex Hausdorff TVS, the closed convex hull of a compact subset is again compact.
The convex hull of compact subset of a
Hilbert space is necessarily closed and so also necessarily compact. For example, let
be the separable Hilbert space
of square-summable sequences with the usual norm
and let
be the standard
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
(that is
at the
-coordinate). The closed set
is compact but its convex hull
is a closed set because
belongs to the closure of
in
but
(since every sequence
is a finite
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other w ...
of elements of
and so is necessarily
in all but finitely many coordinates, which is not true of
). However, like in all complete Hausdorff locally convex spaces, the convex hull
of this compact subset is compact. The vector subspace
is a
pre-Hilbert space when endowed with the substructure that the Hilbert space
induces on it but
is not complete and
(since
). The closed convex hull of
in
(here, "closed" means with respect to
and not to
as before) is equal to
which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might to be compact (although it will be
precompact/totally bounded).
Every complete totally bounded set is relatively compact.
If
is any TVS then the quotient map
is a
closed map and thus
A subset
of a TVS
is totally bounded if and only if its image under the canonical quotient map
is totally bounded. Thus
is totally bounded if and only if
is totally bounded. In any TVS, the closure of a totally bounded subset is again totally bounded.
In a locally convex space, the convex hull and the
disked hull of a totally bounded set is totally bounded. If
is a subset of a TVS
such that every sequence in
has a cluster point in
then
is totally bounded. A subset
of a Hausdorff TVS
is totally bounded if and only if every ultrafilter on
is Cauchy, which happens if and only if it is pre-compact (that is, its closure in the completion of
is compact).
If
is compact, then
and this set is compact. Thus the closure of a compact set is compact
[In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the particular point topology on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. The proof uses and that fact that is compact (but possibly not closed) and is both closed and compact so that which is the image of the compact set under the continuous addition map is also compact. Recall also that the sum of a compact set (that is, ) and a closed set is closed so is closed in ] (that is, all compact sets are
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
). Thus the closure of a compact set is compact. Every relatively compact subset of a Hausdorff TVS is totally bounded.
In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact. More generally, if
is a compact subset of a locally convex space, then the convex hull
(resp. the disked hull
) is compact if and only if it is complete.
Every subset
of
is compact and thus complete.
[Given any open cover of pick any open set from that cover that contains the origin. Since is a neighborhood of the origin, contains and thus contains ] In particular, if
is not Hausdorff then there exist compact complete sets that are not closed.
See also
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Notes
Proofs
Citations
Bibliography
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{{Topological vector spaces
Functional analysis
Topological vector spaces
Uniform spaces