Non-reflexive and non-spherically complete subspaces of the \(p\)-adic space \(\ell^ \infty\). (English) Zbl 0834.46063

Perez-Garcia, C.; Schikhof, W. H.

Indag. Math., New Ser. 6, No. 1, 121-127 (1995).

Summary: By forming tensor products we construct natural examples of non-reflexive and non-spherically complete closed subspaces of the non-Archimedean space \(\ell^\infty\). Also, we study conditions under which two spherically complete Banach spaces are isomorphic; as an application we describe the spherical completion of the subspaces of \(\ell^\infty\) constructed in the paper.

MSC:

46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A45	Sequence spaces (including Köthe sequence spaces)
46M05	Tensor products in functional analysis
46B25	Classical Banach spaces in the general theory

Keywords:

tensor products; non-reflexive; non-spherically complete; non-Archimedean space \(\ell^ \infty\)

References:

[2]	Schikhof, W. H., Locally convex spaces over non-spherically complete valued fields I-II, Bull. Soc. Math. Belgique (ser. B), XXXVIII, 187-224 (1986) · Zbl 0615.46071
[3]	Schikhof, W. H., \(p\)-Adic non-convex compactoids, (Proc. Kon. Ned. Akad. v. Wet., A92 (1989)), 339-342 · Zbl 0712.46045
[4]	Rooij van, A. C.M., Non-archimedean Functional Analysis (1978), Marcel Dekker: Marcel Dekker New York · Zbl 0396.46061