In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ''LB''-space, also written (''LB'')-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 ...
that is a locally convex
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 catego ...
of a countable inductive system
of
Banach spaces.
This means that
is a
direct limit of a direct system
in the category of
locally convex topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s and each
is a Banach space.
If each of the bonding maps
is an embedding of TVSs then the ''LB''-space is called a strict ''LB''-space. This means that the topology induced on
by
is identical to the original topology on
Some authors (e.g. Schaefer) define the term "''LB''-space" to mean "strict ''LB''-space," so when reading mathematical literature, its recommended to always check how ''LB''-space is defined.
Definition
The topology on
can be described by specifying that an absolutely convex subset
is a neighborhood of
if and only if
is an absolutely convex neighborhood of
in
for every
Properties
A strict ''LB''-space is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
,
barrelled, and
bornological (and thus
ultrabornological).
Examples
If
is a locally compact
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 po ...
that is
countable at infinity
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
(that is, it is equal to a countable union of compact subspaces) then the space
of all continuous, complex-valued functions on
with
compact support is a strict ''LB''-space. For any compact subset
let
denote the Banach space of complex-valued functions that are supported by
with the uniform norm and order the family of compact subsets of
by inclusion.
;Final topology on the direct limit of finite-dimensional Euclidean spaces
Let
:
denote the , where
denotes the
space of all real sequences
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real number, real or complex numbers. Equivalently, it is a function space whose elements are functions from the ...
.
For every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
let
denote the usual
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
endowed with the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
and let
denote the canonical inclusion defined by
so that its
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
is
:
and consequently,
:
Endow the set
with the
final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
induced by the family
of all canonical inclusions.
With this topology,
becomes a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
Hausdorff locally convex sequential
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called th ...
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 ...
that is a
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X.
Fréchet–Urysohn spaces are a speci ...
.
The topology
is
strictly finer than the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on
by
where
is endowed with its usual
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...
.
Endow the image
with the final topology induced on it by the
bijection that is, it is endowed with the Euclidean topology transferred to it from
via
This topology on
is equal to the subspace topology induced on it by
A subset
is open (resp. closed) in
if and only if for every
the set
is an open (resp. closed) subset of
The topology
is coherent with family of subspaces
This makes
into an LB-space.
Consequently, if
and
is a sequence in
then
in
if and only if there exists some
such that both
and
are contained in
and
in
Often, for every
the canonical inclusion
is used to identify
with its image
in
explicitly, the elements
and
are identified together.
Under this identification,
becomes a
direct limit of the direct system
where for every
the map
is the canonical inclusion defined by
where there are
trailing zeros.
Counter-examples
There exists a
bornological LB-space whose strong bidual is bornological.
There exists an LB-space 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 ...
.
See also
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Citations
References
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{{TopologicalVectorSpaces
Topological vector spaces