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 quasi-ultrabarrelled space is 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) for which every
bornivorous ultrabarrel is a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of the origin.
Definition
A subset ''B''
0 of a TVS ''X'' is called a bornivorous ultrabarrel if it is a closed,
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
bornivorous subset of ''X'' and if there exists a sequence
of closed balanced and bornivorous subsets of ''X'' such that ''B''
''i''+1 + ''B''
''i''+1 ⊆ ''B''
''i'' for all ''i'' = 0, 1, ....
In this case,
is called a defining sequence for ''B''
0.
A TVS ''X'' is called quasi-ultrabarrelled if every bornivorous ultrabarrel in ''X'' is a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of the origin.
Properties
A
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 ...
quasi-ultrabarrelled space is
quasi-barrelled.
Examples and sufficient conditions
Ultrabarrelled spaces and
ultrabornological space In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spac ...
s are quasi-ultrabarrelled.
Complete and metrizable TVSs are quasi-ultrabarrelled.
See also
*
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 b ...
*
Countably barrelled space In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous.
This property is a generalization of ...
*
Countably quasi-barrelled space In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous.
This property is a generaliza ...
*
Infrabarreled 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 ...
*
Ultrabarrelled space
*
Uniform boundedness principle#Generalisations
References
*
*
*
*
*
*
*
{{Topological vector spaces
Topological vector spaces