veneciasp

2022-07-07

How to solve the following problem:

Let ${f}_{n},f\in {L}^{2}\left({\mathbb{R}}^{d}\right)$ for all $n\ge 1$ be such that $‖{f}_{n}{‖}_{2}\to ‖f{‖}_{2}$ as $n\to \mathrm{\infty }$. Suppose, moreover, that
$\int {f}_{n}g\to \int fg$
for all $g\in {L}^{2}\left({\mathbb{R}}^{d}\right)$. Then ${f}_{n}$ converges to $f$ in ${L}^{2}$-norm.

Oliver Shepherd

Expert

You want to show
$‖f-{f}_{n}{‖}_{2}\to 0$
Equivalently, you can show
$\int \left(f-{f}_{n}{\right)}^{2}d\mu =‖f-{f}_{n}{‖}_{2}^{2}\to 0$
We have
$‖f-{f}_{n}{‖}_{2}^{2}=\int \left(f-{f}_{n}{\right)}^{2}d\mu =\int {f}^{2}d\mu -2\int f{f}_{n}d\mu +\int {f}_{n}^{2}d\mu$
Since we have $‖{f}_{n}{‖}_{2}\to ‖f{‖}_{2}$, we have $\int {f}_{n}^{2}d\mu \to \int {f}^{2}d\mu$ and since we have $\int {f}_{n}gd\mu \to \int fgd\mu$ we have $\int f{f}_{n}d\mu \to \int {f}^{2}d\mu$ so that
$\int {f}^{2}d\mu -2\int f{f}_{n}d\mu +\int {f}_{n}^{2}d\mu \to \int {f}^{2}d\mu -2\int {f}^{2}d\mu +\int {f}^{2}d\mu =0$
that is,
$‖f-{f}_{n}{‖}_{2}^{2}\to 0$
and hence of course also
$‖f-{f}_{n}{‖}_{2}\to 0$

bandikizaui

Expert

Since ${L}^{2}$ is a Hilbert space, you can use the parallelogram identity. More generally, you can also use a property of any uniformly convex Banach space. A very nice proof appears in Brezis' book on functional analysis.