## Real Analysis – How to prove that \$ (C (F), _2) \$ is a pre-hilbertspace space?

Let $$F$$ to be a surface.
For all continuous functions $$f, g in C (F)$$ to define
$$_2: = int_F f (x) g (x) do$$
(integral surface)

I have trouble showing that $$(C (F),<.,.>_2)$$ is a pre-hilbertspace.

Can you help? 🙂 any help very much appreciated