How can I show that $$ f_n=\frac{1}{n}\sum\limits_{k=1}^{n^2} e^{ikt} $$ converges weakly to $0$ in $L^2[-π,π] $ and that the sequence
$$ \left\Vert \frac{1}{n}(f_1+f_2+...+f_n)\right\Vert_2 $$ does not converge to $0$ ?
Thank you in advance.
How can I show that $$ f_n=\frac{1}{n}\sum\limits_{k=1}^{n^2} e^{ikt} $$ converges weakly to $0$ in $L^2[-π,π] $ and that the sequence
$$ \left\Vert \frac{1}{n}(f_1+f_2+...+f_n)\right\Vert_2 $$ does not converge to $0$ ?
Thank you in advance.