Norm of linear functional on product space

Question

Let $\Phi:L^{p}\times L^p\to\mathbb{R}$ be the linear functional given by $$\Phi(h_0,h_1) = \int_If_0h_0+\int_If_1h_1$$ where $f_0,f_1\in L^{q}$, $1/p+1/q=1$. Consider the norm on $L^p\times L^p$ given by $$||(h_0,h_1)|| = ||h_0||_p+||h_1||_p$$ I must prove that $$||\Phi|| = \max\{||f_0||_q,||f_1||_q\}$$ I have shown the inequality $\leq$, I just can't prove the reverse one. I've tried finding an $h$ such that $\Phi$ attains the maximum, but have failed. Any tips?

Answer 1

Consider the functions $$ h_j(x) = \begin{cases} \|f_j\|_q^{1-q} \cdot \frac{|f_j(x)|^{q}}{f_j(x)} & f_j(x) \neq 0\\ 0 & \text{otherwise,} \end{cases} \qquad j=0,1. $$ Notably, $\|h_j\|_p = 1$ for each $j$. It must be true that $|\Phi(h)|$ attains its maximum (over unit $h$) either $(h_0,0)$ or $(0,h_1)$.