Non-reflexive and non-spherically complete subspaces of the p-adic space l∞

Abstract

By forming tensor products we construct natural examples of non-reflexive (Section 2) and nonspherically complete (Section 3) closed subspaces of the non-archimedean space I00. Also, we study (Section 4) conditions under which two spherically complete Banach spaces are isomorphic; as an application we describe the spherical completion of the subspaces of 100 constructed in the paper.

Non-archimedean Eberlein-Šmulian theory

“…E is called normpolar if the norm is polar i.e. if IlxlJ sup{I/(x)l f e E', Ifl -< II} ( e ), in other words, if 3 closed and X C U(x, + u), a U(x + u). COROLLARY 1.6.…”

“…The map f (f(a))ae E is a homeomorphism of (E',w') onto a subspace of K E The image of X lies in the compact subset I-I X(a) so x is w'-precompact. Since E' is W'-aEE quasicomplete by the p-adic Alaoglu Theorem [8], 3.1, it suffices to show that X is w'-closed.…”

On isometries and compact operators between p-adic Banach spaces

“…In the same spirit as Theorem 4.I of [4], the main purpose of this section (Theorem 2.5) is study conditions under which we can assure that E and F are isomorphic, when E and F have orthogonal bases or are spherically complete.…”