Do $L^2$ convergence and continuity imply pointwise convergence? It is said here that $L^2$ convergence and continuity imply pointwise convergence (just before paragraph $5.2$) but I can't find how to prove it. Does anyone see how ?
It's wrong : $f_n(x) = \log(n)e^{-nx}$ over $[0,1]$ is a sequence of continuous functions which converges in $L_2$ toward the zero function.
However, $f_n(0) \rightarrow +\infty$
Convergence in $L^2$ does not imply convergence everywhere but it does imply point wise convergence almost everywhere. See http://en.m.wikipedia.org/wiki/Convergence_of_Fourier_series