The norm of dual operator over $L^p(\Bbb R^N)\times L^p(\Bbb R^N)$

Question

Let $1<p<\infty$ and $E:=L^p(\Bbb R^N)\times L^p(\Bbb R^N)$. Let $\Phi\in E^*$, i.e., the dual of $E$. Hence by Riesz representation we have there exist $u_0$, $u_1\in L^{p'}(\Bbb R^N)$ such that $$ \langle\Phi,(v_0,v_1)\rangle_{E^*,E} = \int_{\Bbb R^N}v_0u_0\,dx+\int_{\Bbb R^N}v_1u_1\,dx $$ for all $(v_0,v_1)\in E$.

Next, my textbook states, without prove, that $$\|\Phi\|_{E^*}=\max\{\|u_0\|_{L^{p'}},\|u_1\|_{L^{p'}}\}$$

Maybe it is an simple fact but I am not very sure how this result is true... Any help is really welcome!!

Answer 1

It is true in general: Let $X_1$, $X_2$ be Banach spaces. And $f_1\in X_1^*$, $f_2\in X_2^*$. If $\Phi : X_1 \times X_2 \to \mathbb F$ is defined by

$$ \Phi(x_1, x_2) = f_1(x_1) + f_2(x_2),$$

then

$$||\Phi|| = \sup_{||x_1||+ ||x_2|| = 1} |f_1(x_1)| + |f_2(x_2)|$$

If you set $x_i = 0$, then you see that $||\Phi|| \geq \max\{ ||f_1||, ||f_2||\}$. On the other hand,

$$|f_1(x_1)| + |f_2(x_2)| \leq ||f_1|| \cdot ||x_1|| + ||f_2|| \cdot ||x_2||$$

as $||f_1|| \cdot ||x_1|| + ||f_2|| \cdot ||x_2|| \leq \max\{ ||f_1||, ||f_2||\}$ whenever $||x_1|| + ||x_2|| = 1$, thus

$$||\Phi|| \leq \max\{ ||f_1||, ||f_2||\}. $$

Thus we are done.