Q:

Inner product of multivariable functions

How is 
$$\langle f, g\rangle \text{where} f(x,y), g(x,y)$$
defined, where
$\langle \cdot ,\cdot \rangle$ means the inner product of $f$ and $g$?
(I primarily searched How can we define the Inner Product of multi-variable functions?)

A:

Inner product for functions (one variable or many variables) is usually defined through definite integrals. One assumes all the functions have the same domain. Take an interval (or a region) in the domain of the function. 
The inner product is the definite integral  over that interval/region of the product of those two functions concerned.

