|
In mathematics, more specifically in functional analysis, a Banach space (pronounced (:ˈbanax)) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced and made a systematic study of them in 1920–1922 along with Hans Hahn and Eduard Helly.〔〕 Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. == Definition == A Banach space is a vector space over the field of real numbers, or over the field of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence in , there exists an element in such that : or equivalently: : The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space is a Banach space if and only if each absolutely convergent series in converges,〔see Theorem 1.3.9, p. 20 in .〕 : Completeness of a normed space is preserved if the given norm is replaced by an equivalent one. All norms on a finite-dimensional vector space are equivalent. Every finite-dimensional normed space over or is a Banach space.〔see Corollary 1.4.18, p. 32 in .〕 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「Banach space」の詳細全文を読む スポンサード リンク
|