Analysis exercises

1. Prove that if $f_n$ converges to $0$ in $L^2$ and $\Vert f_n{\Vert }_{L^2}\le 1$ for each $n$, then $\frac{f_n(x)}{n}$ converges pointwise to $0$ a.e.

Proof: Pick $ϵ>0$. Let $A_n=\{x:\left|\frac{f_n(x)}{n}\right|\ge ϵ\}$. Then we have

$$\begin{array}{rcl}\frac{1}{n^2}& \ge & \int {\left|\frac{f_n}{n}\right|}^2\\ & \ge & {\int }_{A_n}{\left|\frac{f_n}{n}\right|}^2\\ & \ge & {\int }_{A_n}{ϵ}^2\\ & =& {ϵ}^{2}m(A_n)\end{array}$$

so that $m(A_n)\le \frac{1}{{ϵ}^2{n}^2}$. Then $\sum A_n<\mathrm{\infty }$ and so by Borel-Cantelli, $lim sup{A}_n=0$, so for almost all $x$, $lim sup|f_n(x)/n|<ϵ$. Since $ϵ$ was arbitrary, we’re done.