# Math Help - L^2 convergence problems.

1. ## L^2 convergence problems.

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

2. ## Re: L^2 convergence problems.

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

3. ## Re: L^2 convergence problems.

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.

4. ## Re: L^2 convergence problems.

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