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In functional analysis, an F-space is a vector space ''V'' over the real or complex numbers together with a metric ''d'' : ''V'' × ''V'' → R so that # Scalar multiplication in ''V'' is continuous with respect to ''d'' and the standard metric on R or C. # Addition in ''V'' is continuous with respect to ''d''. # The metric is translation-invariant; i.e., ''d''(''x'' + ''a'', ''y'' + ''a'') = ''d''(''x'', ''y'') for all ''x'', ''y'' and ''a'' in ''V'' # The metric space (''V'', ''d'') is complete Some authors call these spaces ''Fréchet spaces'', but usually the term is reserved for locally convex F-spaces. The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be metrizable in a manner that satisfies the above properties. == Examples == Clearly, all Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that .〔Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59〕 The Lp spaces are F-spaces for all and for they are locally convex and thus Fréchet spaces and even Banach spaces. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「F-space」の詳細全文を読む スポンサード リンク
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