Let $H$ be a real reproducing kernel Hilbert space, which can thus be seen as a space of real valued functions defined on some domain $\Omega$. It is known that, if a sequence $\{f_n\}$ of functions of $H$ converges to a certain function $f$ in the norm of the Hilbert space, then $f_n$ converges to $f$ also pointwise. I'd like to know if the inverse holds in a particular case. Explicitly:
Suppose $\{f_n\}$ is a sequence of functions of $H$ such that for all $x\in \Omega$ $$ \lim_{n\to\infty} f_n(x) = f(x) $$ for some function $f\in H$. Suppose moreover that we know that $$ \lim_{n\to\infty} \Vert f - f_n\Vert_H $$ exists. Then is this last limit equal to zero?