1. ## Inner Product Spaces

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?

2. ## Re: Inner Product Spaces

there is nothing wrong with viewing "general" inner products as abstractions based on the euclidean dot product. many kinds of things can be vectors (not just the usual n-tuples), and having a "geometric" tool allows us to have a sense of "spatial intuition" about what is going on.

this is typical of abstraction, in general: we encounter a structure (in this case, inner product vector spaces as exemplified by the real euclidean plane, or euclidean 3-space) and we notice certain aspects of our structure allow us to "simplify calculations". it is natural to ask: what are the rules that "make this structure tick"?

for example, when we are doing algebra with the real or rational numbers, much of the time all we are doing is taking advantage of the field axioms. so a lot of the things we do (such as adding, multiplying, dividing, etc.) would be valid in ANY field. if we can prove something is true for ALL fields F, that certainly saves time re-proving things over and over again, each time we encounter a new field.

the same holds for inner-product spaces. a lot of the things we prove for the "dot product" actually hold for ANY inner product. so why not go "whole hog" and prove things about ALL inner products at once? it's a time-saver.