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An inner product (a generalisation of dot product from high school calculus) on a real vector space V is a scalar-valued function on the Cartesian product of V with itself having the following axioms:

(5.4)

Inner products on complex vector spaces are also possible, but a discussion is outside the scope of this chapter. We also note that inner products are sometimes written as (for example, in physics) instead of where x and y are vectors. For the specific case of n-dimensional vectors we usually write the inner product as follows:

(5.5)

where is the transpose of vector .

This latter notation is common in linear algebra and applications. Of course, more general vector spaces will need the more generic form in axioms (5.4).

An inner product space is a vector space on which an inner product is defined. A finite-dimensional real inner product space is known as a Euclidean space, and a complex inner product space is known as a unitary space. The length of a vector x in Euclidean space is defined to be , and the angle between two vectors x and y is given by:

(5.6)

We say that two vectors x and y are orthogonal if . We immediately see that the zero vector is orthogonal to every other vector. Another example is ; then .

Definition 5.4 The set in an inner product space is orthonormal if:

(5.7)

Finding an orthonormal set in an inner product space is analogous to choosing a set of mutually perpendicular unit vectors in elementary vector analysis.