Difference between revisions of "Ultra-bornological space"
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| − | 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 | + | <!-- |
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| + | 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> | ||
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====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) |
Ultra-bornological space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultra-bornological_space&oldid=17401