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 ...
, 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 al ...
(TVS) is said to be quasi-complete or boundedly complete if every
closed and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
subset is
complete.
This concept is of considerable importance for non-
metrizable TVSs.
Properties
* Every quasi-complete TVS is
sequentially complete.
* 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 ...
space, the closure of the
convex hull 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 "lengt ...
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 ...
TVS then the set of all
compact linear maps of into is a closed vector subspace of
.
* Every quasi-complete
infrabarrelled space 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 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 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 that is not quasi-complete.
See also
*
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References
Bibliography
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{{Topological vector spaces
Functional analysis