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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051170/i0511701.png" /> is infra-barrelled if and only if for any Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051170/i0511702.png" /> every linear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051170/i0511703.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051170/i0511704.png" /> with a closed graph and that maps bounded sets into bounded sets is continuous. See also [[Ultra-barrelled space|Ultra-barrelled space]].
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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 &amp; 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 &amp; Winston  (1965)</TD></TR></table>
 
 
  
 
====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=47358
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article