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