The weak basis theorem for \(\mathbb{K}\)-Banach spaces. (English) Zbl 0786.46059

Kakol, Jerzy

Bull. Soc. Math. Belg., Sér. B 45, No. 1, 1-4 (1993).

Suppose that \(K\) is a non-archimedean spherically complete non-trivially valued field and \(E\) a Banach space over \(K\) with a weak Schauder basis. The author proves that \(E\) is non-archimedean if and only if for every weak Schauder basis \((x_ n)\) in \(E\) the sequence of partial sum operators associated with \((x_ n)\) is pointwise bounded in \(E\). Therefore \(E\) is a non-archimedean space if and only if every weak Schauder basis in \(E\) is a Schauder basis. The above conclusion seems to be useful for some applied problems of functional analysis.

Reviewer: Th.M.Rassias (Athens)

MSC:

46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

non-archimedean spherically complete non-trivially valued field; Schauder basis