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I'm looking for the simpliest counterexample, that bounded sequences in $L^1(\Omega)$ with $|\Omega|<\infty$ may not have weakly convergent subsequence. I'd appreciate if you could at least give me any reference where I can find it.

Thanks in advance, K.

KKK
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1 Answers1

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Take $\Omega = (0,1)$, $$ \phi_n(x) = n \chi_{[0,1/n]}(x), $$ where $\chi_M(x)$ is the characteristic function of the set $M$.

The sequence is bounded, $\|\phi_n\|_{L^1}=1$, but does not converge weakly.

In fact, if the sequence is interpreted as a sequence in the space $C([0,1])^*$, then the sequence converges weak-star towards the Dirac delta at $0$, $\phi_n\rightharpoonup^* \delta_0$ in $C([0,1])^*$.

daw
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