Let $F$ be a surface. For all continuous functions $f,g \in C(F) $ define $$ \langle f,g\rangle_2 := \int_F f(x)g(x)\, dx $$
I'm struggling to show, that $ (C(F),\langle.,.\rangle_2) $ is a pre-hilbert space. Can you help me out? :) Any help will be very appreciated.
Showing that $<f,g>$ is a scalarproduct:
1.
$ 0 \leq $ $<f,f> $ and if $ <f,f>=0 $ $ \leftrightarrow \int_F f(x)f(x) do(x)=0 $
then $f(x) =0 $
2. $<f+h,g>= \int_F (f+h)(x)g(x) do(x) = \int_F f(x)g(x)+\int_f h(x)g(x)= \int_F f(x)g(x) do(x) + \int h(x)g(x) do(x)= <f,g>+<h,g> $
3. $ < \lambda f,g>= \lambda <f,g> $ is clear