Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

up vote 4 down vote accepted

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 $2\lVert f\rVert_{L^2}^2$.

share|improve this answer
    
Simply beautiful. Thanks. –  Romeo Dec 26 '12 at 17:45

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

share|improve this answer
    
Great, thanks for the suggestion, it's very interesting. –  Romeo Dec 26 '12 at 17:45

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.