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An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" (strictly positive and locally finite) measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space; Leonard Gross provided the generalization to the case of a general separable Banach space. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. ==Definition== Let ''H'' be a separable Hilbert space. Let ''E'' be a separable Banach space. Let ''i'' : ''H'' → ''E'' be an injective continuous linear map with dense image (i.e., the closure of ''i''(''H'') in ''E'' is ''E'' itself) that radonifies the canonical Gaussian cylinder set measure ''γ''''H'' on ''H''. Then the triple (''i'', ''H'', ''E'') (or simply ''i'' : ''H'' → ''E'') is called an abstract Wiener space. The measure ''γ'' induced on ''E'' is called the abstract Wiener measure of ''i'' : ''H'' → ''E''. The Hilbert space ''H'' is sometimes called the Cameron–Martin space or reproducing kernel Hilbert space. Some sources (e.g. Bell (2006)) consider ''H'' to be a densely embedded Hilbert subspace of the Banach space ''E'', with ''i'' simply the inclusion of ''H'' into ''E''. There is no loss of generality in taking this "embedded spaces" viewpoint instead of the "different spaces" viewpoint given above. 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「Abstract Wiener space」の詳細全文を読む スポンサード リンク
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