Consider an $L^p(X)$ space, for example a probability space. Suppose that I have a sequence of function $x_n\in X$, such that $\lim_{n\to\infty} \lVert x_n\rVert_p<\infty$. Does this imply that $\{x_n\}_{n}^\infty$ converge to an $x\in X$ under the same norm? If not what are the typical conditions to suffice this?
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https://math.stackexchange.com/questions/138043/does-convergence-in-lp-imply-convergence-almost-everywhere – Tito Eliatron Nov 30 '21 at 18:23
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It does not. A simple example is to look at the standard basis $\{e_n \}_{n \in \mathbb{N}}$ for $\ell^2(\mathbb{N})$. Then $$\lVert e_n\rVert_2 = 1 $$ but the sequence does not converge to anything.
Giorgos Giapitzakis
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Dionel Jaime
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@EdonRon An addition to this answer: Not only that, but ${e_n}$ doesn't even have a convergent subsequence. In particular, not every closed and bounded set is compact. – Just dropped in Nov 30 '21 at 19:12