Example for weak convergence in $L^2(\mathbb{R}^n)$-Norm

Question

We had in the lecture the following example for weak convergence to 0:
$f_k(x) = {k^{-n/2}} f(x/k)$, where $f \in C^\infty (\mathbb{R}^n)$ fixed and $f$ has support in the unit ball $B_1(0)$.
Somehow I don't see why that is true.
I tried to use Hölder but did not get the desired result. I'd be happy to get some hints (I think the $n=1$ case suffices).

Answer 1

You can prove it in the following way. Note that $$\int_{\mathbb{R}^n} |f_k(x)|^2\leq\int_{|x|\le k}k^{-n}\|f\|_\infty\le C$$

where $C$ is a positive constant, hence, we can assume without loss of generality that $f_k$ does cnverge weak to some function $f\in L^2(\mathbb{R}^n)$.

On the other hand, note that for each fixed $x\in\mathbb{R}^n$ we have that $$\lim_{k\to\infty} k^{-n/2}f(x/k)=0,$$

therefore, we have two facts about $f_k$, to wit, it converges to $0$ a.e. and it convergers weakly to some function $f$. To conclude, you can use this post.

Answer 2

Hint: weak konvergence to $0$ means that $\int_{\mathbb R^n} f_k\overline gd\lambda\to 0,\ n\to\infty$ for every $g\in L^2(\mathbb R^n).$ It is enough to consider a dense subset of $g$'s. Take e.g. $g\in C_c(\mathbb R^n)$ - continuous with compact support.