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.
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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