Formally, an inner product space is a real or complex vector space V together with a map f : V x V → F where F is the ground field (either R or C). We write <x, y> instead of f(x, y) and require that the following axioms be satisfied:
A function which follows the second and third axioms is called a sesqui-linear operator (one-and-a-half linear operator). A sesqui-linear operator which is positive (<x, x> ≥ 0) is called a semi inner product.
Here and in the sequel, we will write ||x|| for √<x, x>. This is well defined by axiom 1 and is thought of as the length of the vector x.
From these axioms, we can conclude the following:
An induction on Pythagoras yields:
Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces.
In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V x V to F. This allows us to extend Pythagoras' theorem a tiny bit more, and rename it:
Another consequence of the Cauchy-Schwarz inequality is that it is possible to define the angle φ between two non-zero vectors x and y (at least in the case F = R) by writing
Several types of maps A : V -> W between inner product spaces are of relevance:
From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic.