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Let $(f_n)_n\subseteq L^2(0,1)$ s.t. $$ f_n \rightharpoonup f, \qquad\qquad \Vert f_n\Vert_2 \to \Vert f\Vert_2 $$ where $\rightharpoonup$ means weak convergence. Is it true that $f_n \to f$ strongly, i.e. in $L^2$?

I don't know how to start and I really would like some hints on how to solve it. Actually, I do not know: is it true? I don't manage to find any counterexample...

Is this somehow related to the well-known fact that a.e. convergence + convergence of norms in $L^p$ do imply convergence in $L^p$ (Rudin, R&CA, ex. 17 pag. 73)?

Thanks.

Romeo
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    Can you state more precisely what the "well-known fact" means? We don't all have access to Rudin's books. What does "a.e. convergence + convergence of norms in $L^p$ do imply convergence in L" mean? – Charlie Parker Dec 10 '17 at 19:56
  • I guess I was referring to something of this kind https://math.stackexchange.com/questions/51502/if-f-k-to-f-a-e-and-the-lp-norms-converge-then-f-k-to-f-in-lp (i.e., quite verbatim, if you have a sequence of functions that 1. converges pointwise a.e. and 2.their p-norm make up a convergent sequence (of real numbers), then the sequence converges strongly in $L^p$). – Romeo Dec 25 '17 at 13:10

2 Answers2

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It's almost surely a duplicate, but I think answering is shorter than finding the corresponding one.

Hint: we have $\lVert f-f_n\rVert_{L^2}^2=\lVert f\rVert_{L^2}^2-2\langle f,f_n\rangle+\lVert f_n\rVert_{L^2}^2$. The second term converges to $2\lVert f\rVert_{L^2}^2$.

Davide Giraudo
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In every Uniformly Convex Banach space this is true. See for example the book of Brezis Proposition 3.32 page 78: Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp

Tomás
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