I was reading the book "Functional Analysis" by Brezis and I was searching for an example for a sequence in $L^1(\Omega)$ such that it converges in the weak topology but not in the strong one (the one given by the norm) where $(\Omega, \mathscr{E})$ is a measure space without atoms.
In particular I know that open sets in the weak topology are (strickly) included in the strong one when $L^1$ has infinite dimension. But there are simple examples of a sequence which converges weakly but not strongly in $L^1$ with a general $\Omega$? And in the case $\Omega=\mathbb{R}$?
Best Answer
There is a simple example for $\Omega =\mathbb R$ but a simple example in general may not be possible. Riemann Lebesgue Lemma shows that $\int f(x) g(x) e^{inx} dx \to 0$ as $ n \to \infty$ for all $g \in L^{\infty} (\mathbb R)$ if $f$ is integrable. So $f(x)e^{inx}$ tends to $0$ weakly in $L^{1}$ and it does not tend to $0$ in the norm if $f \neq 0$.