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International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 637-642

Non-archimedean Eberlein-Šmulian theory

1Faculty of Education, Shizuoka University, Ohya, Shizuoka 422, Japan
2Department of Mathematics, University of Nijmegen, Toernooiveld, Nijmegen 6525 ED, The Netherlands

Received 15 December 1994; Revised 8 June 1995

Copyright © 1996 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is shown that, for a large class of non-archimedean normed spaces E, a subset X is weakly compact as soon as f(X) is compact for all fE (Theorem 2.1), a fact that has no analogue in Functional Analysis over the real or complex numbers. As a Corollary we derive a non-archimedean version of the Eberlein-Šmulian Theorem (2.2 and 2.3, for the ‘classical’ theorem, see [1], VIII, §2 Theorem and Corollary, page 219).