Prof Breuer.

I like you answer, I did  this question in order to generate discussion.

Effectively, the answer is yes. Firstly, we need to define a space with an inner product. This is, a topological space X with a function which maps elements of the space on the set of real numbers, complying with some properties. Then, this notion is extended from euclidean spaces to more general spaces. In L2[a,b] the inner product between f and g is defined by the integral of   f g  over [a,b].

Best Regards