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This question asks to show that the following inner product defines a Hilbert space/is complete: $\langle f,g\rangle=\sum_{k=0}^{n}\int_0^1 f^{(k)}(t) \overline{ g^{(k)}}(t)dt$ 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 $\int_0^1 f'\times \overline{g'}$ 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.

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    th$a$nk you $a$lex, and yes that's what I meant.2012-01-16

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