Namespaces
Variants
Actions

Difference between revisions of "Ultra-bornological space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A [[Locally convex space|locally convex space]] which can be represented as an inductive limit of Banach spaces. Alternatively, an ultra-bornological space can be defined as a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950101.png" /> in which every absolutely convex subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950102.png" /> that absorbs each Banach disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950104.png" />, is a neighbourhood of zero. (A Banach disc is an absolutely convex bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950105.png" /> such that its span <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950106.png" /> equipped with the natural norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095010/u0950107.png" /> is a Banach space.) A bornological space is a locally convex space that can be represented as an inductive limit of normed spaces, or, alternatively, a locally convex space in which every absolutely convex subset that absorbs each bounded set, is a neighbourhood of zero.
+
<!--
 +
u0950101.png
 +
$#A+1 = 7 n = 0
 +
$#C+1 = 7 : ~/encyclopedia/old_files/data/U095/U.0905010 Ultra\AAhbornological space
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A [[Locally convex space|locally convex space]] which can be represented as an inductive limit of Banach spaces. Alternatively, an ultra-bornological space can be defined as a locally convex space $  E $
 +
in which every absolutely convex subset $  U $
 +
that absorbs each Banach disc $  A $
 +
in $  E $,  
 +
is a neighbourhood of zero. (A Banach disc is an absolutely convex bounded set $  A $
 +
such that its span $  E _ {A} = \cup _ {n \in \mathbf N }  nA $
 +
equipped with the natural norm $  \| x \| _ {A} = \inf \{ {\rho > 0 } : {x \in \rho A } \} $
 +
is a Banach space.) A bornological space is a locally convex space that can be represented as an inductive limit of normed spaces, or, alternatively, a locally convex space in which every absolutely convex subset that absorbs each bounded set, is a neighbourhood of zero.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Robertson,  W.S. Robertson,  "Topological vector spaces" , Cambridge Univ. Press  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Robertson,  W.S. Robertson,  "Topological vector spaces" , Cambridge Univ. Press  (1964)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:27, 6 June 2020


A locally convex space which can be represented as an inductive limit of Banach spaces. Alternatively, an ultra-bornological space can be defined as a locally convex space $ E $ in which every absolutely convex subset $ U $ that absorbs each Banach disc $ A $ in $ E $, is a neighbourhood of zero. (A Banach disc is an absolutely convex bounded set $ A $ such that its span $ E _ {A} = \cup _ {n \in \mathbf N } nA $ equipped with the natural norm $ \| x \| _ {A} = \inf \{ {\rho > 0 } : {x \in \rho A } \} $ is a Banach space.) A bornological space is a locally convex space that can be represented as an inductive limit of normed spaces, or, alternatively, a locally convex space in which every absolutely convex subset that absorbs each bounded set, is a neighbourhood of zero.

References

[1] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)

Comments

An ultra-bornological space is barrelled and bornological, but the converse is false. Every quasi-complete bornological space is ultra-bornological but, again, the converse fails.

References

[a1] H. Jachow, "Locally convex spaces" , Teubner (1981)
[a2] M. Valdivia, "Topics in locally convex spaces" , North-Holland (1982)
How to Cite This Entry:
Ultra-bornological space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultra-bornological_space&oldid=17401
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article