Convergence of functions in norm and pointwise I understand that $L^2$ convergence does not imply pointwise convergence and vice versa.
But I think that $L^2$ convergence must imply pointwise convergence a. e. $x$?
So, since Schwartz functions are dense in $L^2$, then, for every $f\in L^2$, should it be the sequence $\{S_n\}$ of Schwartz functions that converges to $f$ a.e.?
Please, correct me or approve my understanding. Thank you.
Marina

$L^2$ convergence does not imply pointwise a.e. convergence. One "well-known" example is the sequence
$$ f_1=\chi_{[0,1]},\;f_2=\chi_{[0,\frac{1}{2}]},\; f_3=\chi_{[\frac{1}{2},1]},\; f_4=\chi_{[0,\frac{1}{4}]},\dots $$
in $L^2([0,1])$. This sequence converges to zero in $L^2$, but $\limsup_{n\to\infty}f_n(x)=1$ for all $x\in[0,1]$.
