View source for Ultra-bornological space

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.

====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>



====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Jachow,   "Locally convex spaces" , Teubner  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Valdivia,   "Topics in locally convex spaces" , North-Holland  (1982)</TD></TR></table>