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Functions in non-Archimedean normed fields. (Функции в неархимедовски нормированных полях.) (Russian) Zbl 0293.12105
Saratov: Izdat. Saratov. Univ. 110 p. R. 0.41 (1962).
This is still a useful exposition of some basic results in the theory of functions in non-Archimedean normed (values) fields, with a bibliography upto 1962. An introductory chapter on the subject of normed fields is followed by one on the theory of continuous functions in locally compact non-Archimedean fields and finally the theory of regular functions in complete non-Archimedean fields is described.
{Editorial remark: It has been pointed out a.o. by L. Márki (see his review Zbl 0454.26010), that this book contains an erroneous paragraph, the one on differentiability and Lipschitz functions. The proofs of Theorems 2.6–2.11 are based on a false lemma, and actually, all these theorems are wrong, counter-examples have been given by M. Sh. Stavskii, L. Márki, W. H. Schikhof and J. Mináč.}
Reviewer: B. Vinograde

MSC:
12-02 Research exposition (monographs, survey articles) pertaining to field theory
26-02 Research exposition (monographs, survey articles) pertaining to real functions
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
12J25 Non-Archimedean valued fields
26E30 Non-Archimedean analysis
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis