. As for , the integral is greater than . For the integral is worth 0 so this condition of inner product is satisfied.
Yes, you have to show the 4 properties of inner product to conclude that it is one.
Let V be the space of continuous functions. Show that <f(x), g(x)> = the integral from 0 to 1 of f(x)g(x)dx is an inner product.
I know all of the axioms needed to prove it's an inner product. I'm confused on the definiteness axiom though. For example if it's <v,v>, then it's easy to prove definiteness because it's the same variable; ==> v=0. But in this case, they're two different functions, and I'm not sure how to go about that. And also, is the method for proving the integral as an inner product the same way as proving <f(x), g(x)> is an inner product? Thanks for the help.