Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted and in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space This means that is a linear subspace of the inner product of is the restriction of that of and is dense in for the topology defined by the norm. In this article, F denotes a field that is either the real numbers or the complex numbers A scalar is thus an element of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted for distinguishing it from the scalar 0. An inner product space is a vector space V over the field F together with an inner product, that is, a map that satisfies the following three properties for all vectors and all scalars . Conjugate symmetry: As if and only if is real, conjugate symmetry implies that is always a real number. If F is , conjugate symmetry is just symmetry.