# Sum of inner products in a hilbert space

This question asks to show that the following inner product defines a Hilbert space/is complete: $$\langle f,g\rangle=\sum\int_0^1 f^{(k)}\mathrm{conj}(g^{(k)})$$ where $f^{(k)}(t)$ are continuous on $[0,1]$, $f^{(n-1)}$ is absolutely continuous and $f^{(n)}$ is in $L^2(0,1)$. I've already verified that the space of absolutely continuous functions where $f(0)=0$ and inner product defined by $\int_0^1 f'\times \mathrm{conj}(g)$ is continuous ...this takes it a step further and sums a bunch of inner products that are similar to this one and asks to verify completeness.

I'm not sure how to go about doing this ..

The problem comes from Conway's functional analysis book problem 5 chapter 1 if anyone has the book.

-
I edited your question to use proper formatting. Please check to verify that this is what you were trying to ask. – Alex Becker Jan 16 '12 at 3:08
You have left out some details that would make the problem comprehensible. E.g., what is $k$? What exactly is the set to which this inner product is applied? – Jonas Meyer Jan 16 '12 at 3:09
@JonasMeyer Ah, the sum makes sense if he means $\sum\limits_{k=1}^n$ or something similar. – Alex Becker Jan 16 '12 at 3:14
thank you alex, and yes that's what I meant. – user23024 Jan 16 '12 at 3:21