Let $x_{n}$ be a seuquence in $\mathbb{R}$, define the sequence of functions: $$u_{n}=\chi_{B(x_{n},1)}$$ where $\chi$ denotes the characteristic function. Study the convergence of $u_{n}\in L^{2}(\mathbb{R})$ in the strong topology and weak topology in the following cases cases:
a) $x_{n}\longrightarrow0$
b)$|x_{n}|\longrightarrow+\infty$
The first case should be the easier, since we expect the limit to be $0$, if it exists. So we may compute: $$\int_{\mathbb{R}}(\chi_{B(x_{n},1)})^{2}dx=\int_{x_{n}-1}^{x_{n}+1}dx=2$$ Thus there can't be strong convergence. Regarding weak convergence, we know $$u_{n}\rightharpoonup0\iff\int_{\mathbb{R}}u_{n}\varphi dx\longrightarrow0\quad\forall\varphi\in L^{2}(\mathbb{R}) $$ so why not $\varphi=u_{n}$ again. Then we would have $u_{n}$ doesn't converge weakly in this case. Does this idea work? Am I missing something? Does case $b)$ follow similarly?
