Inner Product Space - Examples

Examples

• A simple example is the real numbers with the standard multiplication as the inner product

More generally, any Euclidean space n with the dot product is an inner product space

where xT is the transpose of x.

• The general form of an inner product on n is known as the Hermitian form and is given by

where M is any Hermitian positive-definite matrix, and x* the conjugate transpose of x. For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.

• The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C of continuous complex valued functions on the interval . The inner product is

This space is not complete; consider for example, for the interval the sequence of "step" functions { f_k }_k where

• f_k(t) is 0 for t in the subinterval
• f_k(t) is 1 for t in the subinterval
• f_k is affine in (0, 1/k). That is, f_k(t) = kt.

This sequence is a Cauchy sequence which does not converge to a continuous function.

• For random variables X and Y, the expected value of their product

is an inner product. In this case, <X, X>=0 if and only if Pr(X=0)=1 (i.e., X=0 almost surely). This definition of expectation as inner product can be extended to random vectors as well.

• For square real matrices, with transpose as conjugation is an inner product.