Suppose that $\Omega$ is a Lebesgue measurable set，$f_n \rightharpoonup f$ in $L ^1(\Omega)$ and $\|f_n\|_{L^1(\Omega)}\rightharpoonup\|f\|_{L^1(\Omega)}$, then can I say that $f_n → f$ strongly in $L^1(\Omega)$? And how to prove it if it is true?
2026-03-30 03:19:55.1774840795
Weak convergence and strong convergence in $L^1$.
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I remember that this statement is true in Hilbert Spaces and for $1<p<\infty$, I think for the case $p=1$ it does not hold as $L^1$ is not reflexive. You should be able to construct a counterexample using a Dirac-Sequence.