The three-space problem for a class of normed spaces. (English) Zbl 0629.46068

Martinez-Maurica, J.; Pérez Garcia, C.

Bull. Soc. Math. Belg., Sér. B 39, 209-214 (1987).

It is known [N. J. Kalton, Compos. math. 37, 243-276 (1978; Zbl 0395.46003); M. Ribe, Proc. Am. Math. Soc. 73, 351-355 (1979; Zbl 0397.46002); J. W. Roberts, Lect. Notes Math. 604, 76-81 (1977; Zbl 0369.46009)] that if X is an F-space with a closed subspace M for which both it and X/M are locally convex, then X need not be locally convex. The authors shows that for complete locally bounded non-archimedean spaces over a spherically complete valued field this pathology cannot arise. This is done by first showing that in this setting every Banach space is a K-space.

Reviewer: A.C.Thompson

MSC:

46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A04	Locally convex Fréchet spaces and (DF)-spaces
46B20	Geometry and structure of normed linear spaces

Keywords:

three-space problem; F-space; complete locally bounded non-archimedean spaces over a spherically complete valued field; K-space

Citations:

Zbl 0395.46003; Zbl 0397.46002; Zbl 0369.46009