# Math Help - The pointwise limit of an orthonormal sequence

1. ## [Solved]limit of ansequence

solved, thank you all

2. Suppose, for a contradiction, that there is a set of positive measure on which f is not zero. Then by Egorov's theorem there is a (maybe smaller) set A on which $f_n(x)\to f(x)$ uniformly. Let g be the function that is equal to f on A and 0 outside A. Then $\int_A|g|^2d\mu>0$.

On the other hand, it follows from Parseval's equality that $\langle g,f_n\rangle\to0$ as n→∞. But $\langle g,f_n\rangle = \int_X g\bar{f_n}\,d\mu = \int_A g\bar{f_n}\,d\mu$ since g is zero off A. This converges to $\int_A |g|^2d\mu$ as n→∞, by the uniform convergence. That gives your contradiction.

3. thank you for your help!