My aim is to prove that the space: $\mathcal{H}$ = {$f:[0,1] \to \mathbb{R}: f\;is\;absolutely\;continuous,\;f(0)=f(1)=0,\;f'\in L^2[0,1]$} is a reproducing kernel hilbert space.
Now assuming an inner-product given by: $<f,g> = \int_0^1\;f'(t)g'(t)\;dt$, by writing $f(x) = \int_0^x f'(t)\;dt$
I am able to show using cauchy schwartz inequality that $|f(x)|\leq ||f|| \sqrt{x}$ and hence show that the Point evaluation functional is bounded. However, for completing the proof I have to show that this space is complete (So that it becomes a hilbert space). How do I do this?