Representing a bounded linear functional on $L^p$

Question

I have a question on this problem, and note it is for homework.

Let $p,q$ be conjugate exponents. Define $H\colon L^p([0,1])\to \mathbb{R}$ by $$ Hf=2\int_0^1\left(\int_x^1 f(y)\,dy\right) x\,dx. $$ Find a function $g\in L^q([0,1])$ so that $$ Hf=\int_0^1fg \,dx. $$

Riesz representation theorem comes to mind, but I'm not sure how to apply it.

Answer 1

It's Fubini Theorem indeed:

$$2\int_0^1\left(\int_x^1 f(y)\,dy\right) x\,dx = 2\int_0^1 \left(\int_0^y x f(y) \,dx\right) \,dy = \int_0^1 y^2 f(y) dy.$$

Thus

$$Hf = \int_0^1 y^2 f(y) \,dy$$

and so $g(x) = x^2$.