The complex absolute value is a special case of the norm in an inner product space.

Let A be an operator on a finite-dimensional inner product space.

The general result for an inner product space was obtained by Schwarz in the year 1885.

Any complete inner product space V has an orthonormal basis.

The inner product space can be identified naturally with Ran(D).

Every quadratic space is also an inner product space (see below).

A Hilbert space is defined as a complete inner product space.

However, the result will only be an inner product space and it will not necessarily be complete.

Every finite-dimensional inner product space is also a Hilbert space.

This formula is valid for any inner product space, including Euclidean and complex spaces.

Angles between vectors are defined in inner product spaces.

Let V be any inner product space.

Let 'V' be a finite dimensional inner product space of dimension 'n'.

Parseval's identity: Suppose V is a complete inner product space.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis.

The set, together with the specific inner product specify the inner product space.

It is defined on the exterior algebra of a finite-dimensional oriented inner product space.

It is the standard 'inner product space' of the orthonormal Euclidean space.

They are all special cases of decompositions over an orthonormal basis of an inner product space.

The isosceles triangle theorem holds in inner product spaces over the real or complex numbers.

This means that at each point of M, the fibre of E is an inner product space.

Inner product spaces also define an operation known as the inner product.

All -dimensional real inner product spaces are mutually isomorphic.

Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.

Mercer's theorem asserts that can then be expressed as with mapping the arguments into an inner product space.