# Show an operator on L^2(0,\infty) is bounded

1. Nov 12, 2011

### economist13

1. The problem statement, all variables and given/known data

Show that the operator on $$L^2(0,\infty)$$ defined by $$g \rightarrow f(x)= \int_{0}^{\infty} e^{-xy}g(y)dy$$ is bounded.

2. Relevant equations

Operator norm: $$||T|| = \sup_{||g||_{L^2}=1}||Tg||_{L^2}$$

3. The attempt at a solution

I tried to get a handle on $$f(x)= \int_{0}^{\infty} e^{-xy}g(y)dy$$, by first fixing x and applying Holder's inequality. I got that for every x, $$f(x) \leq \frac{1}{2x}$$ but this didn't really get me much since $$\int_{0}^{\infty}(\frac{1}{2x})^2dx$$ doesn't converge...

What I need is that $$\int_{0}^{\infty} | \int_{0}^{\infty}e^{-xy}g(y)dy|^2dx$$ is finite. any inequalities/tricks i'm not thinking of? I'm hoping I'm just not realizing what theorem/inequality I need to use....I really appreciate any help you all can offer