# Square integrable functions

1. Nov 18, 2009

### NSAC

Hi i have a question about $L^2$ spaces and convergence.
Here it goes:
Let $K\subset \mathbb{R}^2$ be bounded.
Let $g,h\in L^2(K)$, and a sequence $f_n\in L^2(K)$ such that $f_n$ converges strongly to $f\in L^2$.
Is it true that $\lim_{n\rightarrow \infty} \int_{K} f_n g h = \int_{K} f g h$? If it is how?
Thank you.

Last edited by a moderator: Dec 2, 2009
2. Nov 19, 2009

### zhangzujin

$g,h \in L^2$, then $gh\in L^1$ by Holder inequality.

and so I do not know the integral $\int_K fgh$ is well-defined?

Last edited by a moderator: Dec 2, 2009
3. Dec 2, 2009

### NSAC

I didn't get what you mean. Are you asking it as a question?

4. Dec 2, 2009

### HallsofIvy

Staff Emeritus
Yes. $L^2$ consists of functions whose square is integrable. The product of any two such functions is integrable but the product of three of them may not be.
($L^2$ is not closed under multiplication.)