Do weak convergence and convergence of norms imply convergence in $L^1$?

Question

I know this does hold in $L^2$, since it's a Hilbert space.

I suspect that this is not true, but I cannot think of a counterexample.

Specifically, I want to know if $f_n \xrightarrow{w} f$ and $\Vert f_n \Vert \rightarrow \Vert f \Vert$ implies $\Vert f_n -f \Vert \rightarrow 0$.

Answer 1

It $\textit{is}$ true in $L^2:\ \|f-f_n\|^2=\|f\|^2-2\langle f,f_n\rangle+\|f_n\|^2\to 0$ as $n\to \infty.$ In fact, the result is true for $f\in L^p:\ 1<p<\infty$ because these spaces are reflexive.

For a nice counterexample in $L^1([0,2])$, see the comment here.