Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty \frac{f^{(n)}(0)g^{(n)}(0)}{(n!)^2} \hspace{3in}$$ The following function $f$ poses problems:
$f(x) = \left\{ \begin{array}{ll} e^{-1/x^2} & \mbox{if } x \neq 0\\ 0 & \mbox{if } x = 0 \end{array} \right.$
This is because $f^{(n)}(0) = 0$ for all $n \in \mathbb{N}$. So $<f,f> = 0$ but $f \neq 0$. Is there a natural subspace of infinitely differentiable functions that is larger than the polynomials where this makes a good inner product?