Ultra-bornological space

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)