Proving a linear form to be continuous

Question

In the Hilbert space $L^2(R)$ I have seen that the following form is linear, however, I need to check if it is continuous and find the associated vector using the Riesz-Fréche Theorem. I have tried to prove that it is bounded but I was not able to do it.

$$F(f) = \int_0^1 f(x^2)xdx$$

Answer 1

You can simply make a change of variable $y=x^2$. So $$F(f) = \frac{1}{2}\int_0^1 f(y)dy$$ which is clearly bounded. The constant function $1/2$ is the relative dual element in Riesz-Fréchet.