The bidual of a non-archimedean locally convex space. (English) Zbl 0693.46071

de Grande-de Kimpe, N.

Indag. Math. 51, No. 2, 203-212 (1989).

Let K be a complete nontrivially valued ultrametric field and let E be a locally convex Hausdorff space over K whose topological dual space \(E'\) separates the points of E. The space E is said to be semi-reflexive if the canonical (injective) mapping \(J_ E\) from E into \(E''\) is surjective.
If (E,F) is a dual pair of spaces, \(\sigma\) (E,F) denotes a strong dual pair topology on E.
The author shows that E is semi-reflexive if and only if it is \(\sigma (E,E')\)-quasi-complete. Moreover she shows that if E is \(\sigma (E,E')\)- quasi-complete then E is quasi-complete hence every semi-reflexive space is quasi-complete. She also obtains several sufficient conditions to have E sequentially complete.
If E is semi-reflexive and has the Hahn-Banach extension property then she shows that every closed subspace is semi-reflexive too. But she gives an example of non semi-reflexive quotient of E by a closed subspace.
When K is not semi-reflexive she shows that every quasi-complete space of countable type is semi-reflexive.
When E is a polar space she obtains results linked to the property “E is p-bornological”. In particular if E is metrizable, then it is p- bornological.

Reviewer: A.Escassut

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:

ultrametric Banach spaces; nontrivially valued ultrametric field; locally convex Hausdorff space; semi-reflexive; dual pair of spaces; quasi- complete; Hahn-Banach extension property; quasi-complete space of countable type; polar space; p-bornological