TITLE: How to motivate inner product spaces
QUESTION [5 upvotes]: What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of general vectors spaces such as the set of matrices, polynomials, functions?

REPLY [2 votes]: As mentioned in the comments it's hard to go wrong with Fourier Polynomials. 
But there are all sorts of special polynomials (with applications galore) which are (or at least can be) defined in terms of orthogonality...for example: Laguerre polynomials, 
Hermite polynomial, and 
Chebyshev polynomials.