Let us consider the space $L^1$ of measurable functions associated to a probability space. I would like to see non trivial examples of weakly-compact sets.
Thank you.
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In light of the Dunford-Pettis theorem you are really just asking about interesting examples of weakly closed, uniformly integrable families. Adding the tag. – Nate Eldredge Sep 09 '15 at 13:53
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Fix $g\in L^1$ non-negative and consider $$ K:= \{f \in L^1 : |f| \leq g\} $$ Then $K$ is bounded by $\|g\|_1$ and uniformly integrable (by the absolute continuity of $\int g$), and so it is weakly compact by the Dunford-Pettis theorem. (See this)
Edit: As pointed out by Norbert, it remains to show that $K$ is weakly closed. Since it is convex, it suffices to show that it is strongly closed. However, if $(f_n) \subset K$ with $f_n \to f$ in $L^1$, then there is a subsequence $(f_{n_k})$ which converges pointwise to $f$. From this it follows that $|f| \leq g$ holds as well.
Prahlad Vaidyanathan
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