﻿

# inner product space

In mathematics, a vector space or function space in which an operation for combining two vectors or functions (whose result is called an inner product) is defined and has certain properties.

Such spaces, an essential tool of functional analysis and vector theory, allow analysis of classes of functions rather than individual functions. In mathematical analysis, an inner product space of particular importance is a Hilbert space, a generalization of ordinary space to an infinite number of dimensions. A point in a Hilbert space can be represented as an infinite sequence of coordinates or as a vector with infinitely many components. The inner product of two such vectors is the sum of the products of corresponding coordinates. When such an inner product is zero, the vectors are said to be orthogonal (see orthogonality). Hilbert spaces are an essential tool of mathematical physics. See also David Hilbert.

* * *

Universalium. 2010.

### Look at other dictionaries:

• Inner product space — In mathematics, an inner product space is a vector space with the additional structure of 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.… …   Wikipedia

• Indefinite inner product space — In mathematics, in the field of functional analysis, an indefinite inner product space :(K, langle cdot,,cdot angle, J) is an infinite dimensional complex vector space K equipped with both an indefinite inner product :langle cdot,,cdot angle and… …   Wikipedia

• inner product — Math. 1. Also called dot product, scalar product. the quantity obtained by multiplying the corresponding coordinates of each of two vectors and adding the products, equal to the product of the magnitudes of the vectors and the cosine of the angle …   Universalium

• Petersson inner product — In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson. DefinitionLet mathbb{M} k be the space of entire modular forms of weight k… …   Wikipedia

• Inner space — may refer to:* Inner Space (album), a 1966 jazz music album by Chick Corea * Inner Space, A conference of closed circuit rebreather diving and CCR training held annually in the Grand Caymans. * University of Rhode Island, Inner Space Center *… …   Wikipedia

• Space (mathematics) — This article is about mathematical structures called spaces. For space as a geometric concept, see Euclidean space. For all other uses, see space (disambiguation). A hierarchy of mathematical spaces: The inner product induces a norm. The norm… …   Wikipedia

• space — 1. noun /speɪs/ a) The intervening contents of a volume. If it be only a Single Letter or two that drops, he thruſts the end of his Bodkin between every Letter of that Word, till he comes to a Space: and then perhaps by forcing thoſe Letters… …   Wiktionary

• inner — innerly, adv., adj. innerness, n. /in euhr/, adj. 1. situated within or farther within; interior: an inner door. 2. more intimate, private, or secret: the inner workings of the organization. 3. of or pertaining to the mind or spirit; mental;… …   Universalium

• space — spacer, n. /spays/, n., v., spaced, spacing, adj. n. 1. the unlimited or incalculably great three dimensional realm or expanse in which all material objects are located and all events occur. 2. the portion or extent of this in a given instance;… …   Universalium

• product — /prod euhkt, ukt/, n. 1. a thing produced by labor: products of farm and factory; the product of his thought. 2. a person or thing produced by or resulting from a process, as a natural, social, or historical one; result: He is a product of his… …   Universalium