Is $u \mapsto \|u\|_p$ a bounded linear functional on $L^p(\Omega)$?

Question

Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $1<p<\infty$. My question: is $F:L^p(\Omega) \rightarrow [0,\infty)$, $F(u)=\|u\|_p$ a bounded linear functional? I would say that it is, with $\|F\|=1$, but from that would follow that there exists $h\in L^q(\Omega)$ such that $$F(u) = \|u\|_p = \int_{\Omega} h(x) u(x) \ d\mu(x) \quad \text{ for all }u\in L^P(\Omega).$$ (obviously $\frac{1}{p}+\frac{1}{q}=1$ and $\mu$ is the Lebesgue measure) If this really is true... it is quite shocking for me. Could you please clarify or verify?

Answer 1

The function you're describing isn't linear. Instead, we have (in)equalities like $\|\alpha f\| = |\alpha|\|f\|$ and $\|u+v\| \le \|u\| + \|v\|$ - we don't have $F(\alpha u+v) = \alpha F(u) + F(v)$.