Is this space a Hilbert Space? 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.

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.
