I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way:
$ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $
Is this space a Hilbert Space? I think that completness of the space should be checked, but i don't know how to do it.
Comparing with the space of continuous functions on [a, b] (not mandatory differentiable) which has the dot product $ x \cdot y = \int_a^b \! x(t)y(t) \, \mathrm{d}t $ i see that my space and dot product (with derivatives) exclude some standart functional sequences that help to prove that the space of continuous functions is incomplete. I mean that, for example, this functional sequence $ f_n(t) =
\begin{cases}
-1, & \text{if }t\text{ in [-1, -1/n]} \\
nt, & \text{if }t\text{ in [-1/n, 1/n]} \\
1, & \text{if }t\text{ in [1/n, 1]}
\end{cases} $ shows that the space of continuous functions is incomplete, but it is not appliable to my problem, because it is not continuously differentiable.
Best Answer
You're getting close with your example functions $f_n$ on $[-1,1]$. Try letting $g_n(t) = \int_{-1}^t f_n(s)\,ds$. Show $g_n$ converges in your norm to a function which is not continuously differentiable.