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Nonarchimedean Normed Linear Spaces
Author(s) -
REN GUANSHEN
Publication year - 1992
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1992.tb32259.x
Subject(s) - mathematics , dedekind cut , banach space , completeness (order theory) , lemma (botany) , valuation (finance) , rank (graph theory) , pure mathematics , normed algebra , normed vector space , discrete mathematics , algebra over a field , combinatorics , mathematical analysis , division algebra , ecology , poaceae , finance , economics , filtered algebra , biology
Important theorems of valuation theory and classic functional analysis that have known analogues in the theory of nonarchimedean normed linear spaces over rank‐one valued fields are considered in the more general context of nonarchimedean normed linear spaces over fields of arbitrary rank. Maximal completeness, pseudocompleteness, and spherical completeness are shown to be equivalent. Versions of Riesz's lemma and the Hahn–Banach theorem are given. The utility of considering ordered (value) groups as being embedded in their Dedekind completions is demonstrated.