L^2 convergence problems.

**younhock** · December 6th 2011, 06:40 AM

Suppose $f_n,f,g \in L^2 (A)$ and $f_n \rightarrow f in L^2$ . Is is true if i say $lim_{n\rightarrow \infty} \int_A f_n g dm = \int_A f g dm$ ?

**girdav** · December 6th 2011, 07:40 AM

Apply Cauchy-Schwarz inequality to see it's true.

**younhock** · December 6th 2011, 07:51 AM

Originally Posted by girdav

Apply Cauchy-Schwarz inequality to see it's true.

Sorry, can you tell me more detail, because i cant prove it. THanks very much.

**girdav** · December 6th 2011, 07:56 AM

$\left|\int_Af_ng-\int_Afg\right|\leq \int_A |f-f_n||g|\leq \sqrt{\int_A |f-f_n|^2}\sqrt{\int_A g^2}$ , and you can conclude (the hypothesis is the strong convergence, whereas the conclusion is called weak convergence).

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