Why is the dimension of the Hilbert space of square-integrable functions infinite? I guess I'm having trouble visualizing the basis of this space.
Why not just show a concrete infinite set of linearly-independent square-integrable functions? (Although, come to think of it, we must know something more about the domain of those square-integrable functions. If that domain can be any measure space, then the proposition might not even be true in general.)
For example, if are talking about the space of square integrable functions on a finite interval, then all functions of theform for positive integer n are in the space. Can you show that they are independent?
A set of functions like sin(nx) for integer n would be a natural choice also.