翻訳と辞書 Words near each other ・ Inner moon ・ Inner Motion ・ Inner Ninja ・ Inner North ・ Inner North Coast Ranges ・ Inner nuclear layer ・ Inner nuclear membrane protein ・ Inner painting ・ Inner Party ・ Inner path ・ Inner peace ・ Inner Pierhead Fort ・ Inner Plane ・ Inner plexiform layer ・ Inner Port Shelter ・ Inner product space ・ Inner Reef ・ Inner regular measure ・ Inner Relationship Focusing ・ Inner Revolution ・ Inner Ring Road ・ Inner Ring Road (Shanghai) ・ Inner Ring Road (Tianjin) ・ Inner Ring Road Multan ・ Inner Ring Road, Bangalore ・ Inner Ring Road, Berlin ・ Inner Ring Road, Chennai ・ Inner Ring Road, Hyderabad ・ Inner Rocks ・ Inner root sheath Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Inner product space ： ウィキペディア英語版 Inner product space In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An incomplete space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. == Definition == In this article, the field of scalars denoted is either the field of real numbers or the field of complex numbers . Formally, an inner product space is a vector space over the field together with an ''inner product'', i.e., with a map :$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; V\; \backslash times\; V\; \backslash to\; F$ that satisfies the following three axioms for all vectors $x,y,z\; \backslash in\; V$ and all scalars $a\; \backslash in\; F$: * Conjugate symmetry:〔A bar over an expression denotes complex conjugation.〕 ::$\backslash langle\; x,y\backslash rangle\; =\backslash overline$ * Linearity in the first argument: ::$\backslash langle\; ax,y\backslash rangle\; =\; a\; \backslash langle\; x,y\backslash rangle$ ::$\backslash langle\; x+y,z\backslash rangle\; =\; \backslash langle\; x,z\backslash rangle\; +\; \backslash langle\; y,z\backslash rangle$ * Positive-definiteness: ::$\backslash langle\; x,x\backslash rangle\; \backslash geq\; 0$ ::$\backslash langle\; x,x\backslash rangle\; =\; 0\; \backslash Rightarrow\; x\; =\; \backslash mathbf$ :If the second condition in positive-definiteness is dropped, the resulting structure is called a semi-inner product. 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「Inner product space」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.