Inner product space

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In mathematics, an inner product space is a vector space with the 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 space is sometimes also called a pre-Hilbert space, since its completion with respect to the metric induced by its inner product, is a Hilbert space. That is, if a pre-Hilbert space is complete with respect to the metric arising from its inner product (and norm), then it is called a Hilbert space.

Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used.

Contents

Definition

In this article, the field of scalars denoted \mathbb{F} is either the field of real numbers \mathbb{R} or the field of complex numbers \mathbb{C}.

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