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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 : that satisfies the following three axioms for all vectors and all scalars : * Conjugate symmetry:〔A bar over an expression denotes complex conjugation.〕 :: * Linearity in the first argument: :: :: * Positive-definiteness: :: :: :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
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