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 ...
, 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) is said to be quasi-complete or boundedly complete if every
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded subset 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 ...
.
This concept is of considerable importance for non-
metrizable TVSs.
Properties
* Every quasi-complete TVS is
sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in .
is called sequentially complete if i ...
.
* In a quasi-complete
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 ...
space, the closure of the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of a compact subset is again compact.
* In a quasi-complete Hausdorff TVS, every
precompact subset is relatively compact.
* If is a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
and is a quasi-complete
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 ...
TVS then the set of all
compact linear map
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
s of into is a closed vector subspace of
.
* Every quasi-complete
infrabarrelled space
In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infra barreled) if every bounded absorbing barrel is a neighborhood of the origin.
Characteri ...
is barreled.
* If is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is
strongly bounded.
* A quasi-complete
nuclear space
In mathematics, nuclear spaces are topological vector space, topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite diff ...
then has the
Heine–Borel property.
Examples and sufficient conditions
Every complete TVS is quasi-complete.
The product of any collection of quasi-complete spaces is again quasi-complete.
The projective limit of any collection of quasi-complete spaces is again quasi-complete.
Every
semi-reflexive space In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual ...
is quasi-complete.
The quotient of a quasi-complete space by a closed vector subspace may ''fail'' to be quasi-complete.
Counter-examples
There exists an
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 sys ...
that is not quasi-complete.
See also
*
*
References
Bibliography
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{{Topological vector spaces
Functional analysis