Difference between revisions of "Infra-barrelled space"
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A locally convex [[Linear topological space|linear topological space]] in which every barrel (i.e. absorbing convex closed [[Balanced set|balanced set]]) absorbing any bounded set is a neighbourhood of zero. The barrelled spaces form an important class of infra-barrelled spaces (cf. [[Barrelled space|Barrelled space]]). | A locally convex [[Linear topological space|linear topological space]] in which every barrel (i.e. absorbing convex closed [[Balanced set|balanced set]]) absorbing any bounded set is a neighbourhood of zero. The barrelled spaces form an important class of infra-barrelled spaces (cf. [[Barrelled space|Barrelled space]]). | ||
| − | Inductive limits and arbitrary products of infra-barrelled spaces are infra-barrelled. A space is infra-barrelled if and only if either every bounded lower semi-continuous [[Semi-norm|semi-norm]] is continuous, or if every strongly-bounded subset in the dual space (cf. [[Adjoint space|Adjoint space]]) is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]). In particular, every bornological space (i.e. a space in which every bounded semi-norm is continuous) is an infra-barrelled space. In a sequentially-complete linear topological space infra-barrelledness implies barrelledness; infra-barrelled spaces, like barrelled spaces, can be characterized in terms of mappings into Banach spaces: A locally convex linear topological space | + | Inductive limits and arbitrary products of infra-barrelled spaces are infra-barrelled. A space is infra-barrelled if and only if either every bounded lower semi-continuous [[Semi-norm|semi-norm]] is continuous, or if every strongly-bounded subset in the dual space (cf. [[Adjoint space|Adjoint space]]) is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]). In particular, every bornological space (i.e. a space in which every bounded semi-norm is continuous) is an infra-barrelled space. In a sequentially-complete linear topological space infra-barrelledness implies barrelledness; infra-barrelled spaces, like barrelled spaces, can be characterized in terms of mappings into Banach spaces: A locally convex linear topological space $ X $ |
| + | is infra-barrelled if and only if for any Banach space $ Y $ | ||
| + | every linear mapping from $ X $ | ||
| + | into $ Y $ | ||
| + | with a closed graph and that maps bounded sets into bounded sets is continuous. See also [[Ultra-barrelled space|Ultra-barrelled space]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 22:12, 5 June 2020
A locally convex linear topological space in which every barrel (i.e. absorbing convex closed balanced set) absorbing any bounded set is a neighbourhood of zero. The barrelled spaces form an important class of infra-barrelled spaces (cf. Barrelled space).
Inductive limits and arbitrary products of infra-barrelled spaces are infra-barrelled. A space is infra-barrelled if and only if either every bounded lower semi-continuous semi-norm is continuous, or if every strongly-bounded subset in the dual space (cf. Adjoint space) is equicontinuous (cf. Equicontinuity). In particular, every bornological space (i.e. a space in which every bounded semi-norm is continuous) is an infra-barrelled space. In a sequentially-complete linear topological space infra-barrelledness implies barrelledness; infra-barrelled spaces, like barrelled spaces, can be characterized in terms of mappings into Banach spaces: A locally convex linear topological space $ X $ is infra-barrelled if and only if for any Banach space $ Y $ every linear mapping from $ X $ into $ Y $ with a closed graph and that maps bounded sets into bounded sets is continuous. See also Ultra-barrelled space.
References
| [1] | N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) |
| [2] | R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) |
Comments
"Infra-barrelled spaces" are also called quasi-barrelled spaces.
References
| [a1] | H. Janchow, "Locally convex spaces" , Teubner (1981) |
How to Cite This Entry:
Infra-barrelled space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infra-barrelled_space&oldid=16184
Infra-barrelled space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infra-barrelled_space&oldid=16184
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article