The Mackey-Arens property for spaces over valued fields. (English) Zbl 0819.46062

Kąkol, Jerzy

Bull. Pol. Acad. Sci., Math. 42, No. 2, 97-101 (1994).

Summary: We show that if \(F\) is a spherically complete normed space over a non- Archimedean complete non-trivially valued field \(\mathbb{K}\), then \(F\) has the Mackey-Arens property, i.e. every locally convex space \(E\) over \(\mathbb{K}\) admits the finest locally convex topology \(\mu\) such that \(L((E, \mu), F)= L(E,F)\). We show also that in general the Mackey-Arens property does not characterize the spherical completeness of normed complete spaces, although (as we show) it does for many concrete spaces. This extends a result of J. Van Tiel [Indag. Math. 27, 249-289 (1965; Zbl 0133.065)].

MSC:

46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A20	Duality theory for topological vector spaces

Keywords:

spherically complete normed space over a non-Archimedean complete non- trivially valued field; Mackey-Arens property

Citations:

Zbl 0133.065