1. ## L^2 integral problem.

Suppose that $\displaystyle fg \in L^2 ([a,b])$ for all $\displaystyle f \in L^2([a,b])$. How to show that $\displaystyle g$ is also in $\displaystyle L^2([a,b])$

2. ## Re: L^2 integral problem.

Take f(x) = 1.

3. ## Re: L^2 integral problem.

Originally Posted by emakarov
Take f(x) = 1.
this doesn't show for every f.

4. ## Re: L^2 integral problem.

Originally Posted by younhock
Suppose that $\displaystyle fg \in L^2 ([a,b])$ for all $\displaystyle f \in L^2([a,b])$. How to show that $\displaystyle g$ is also in $\displaystyle L^2([a,b])$
I thought you assume that $\displaystyle fg \in L^2 ([a,b])$ for all $\displaystyle f \in L^2([a,b])$. We have $\displaystyle 1 \in L^2([a,b])$.

5. ## Re: L^2 integral problem.

Originally Posted by emakarov
I thought you assume that $\displaystyle fg \in L^2 ([a,b])$ for all $\displaystyle f \in L^2([a,b])$. We have $\displaystyle 1 \in L^2([a,b])$.
Sorry I not really get what you mean. Would you explain in details?

6. ## Re: L^2 integral problem.

You are given that $\displaystyle fg \in L^2 ([a,b])$ for all $\displaystyle f \in L^2 ([a,b])$. This is known; you don't have to prove this. Therefore, you can replace $\displaystyle f$ with any function, as long as it is in $\displaystyle L^2 ([a,b])$, and still get a true statement. I suggest replacing $\displaystyle f$ with a function that equals 1 everywhere.