I have the following operator:
$\langle Au, v\rangle = \int_{\mathbb{R}^d} |u(x)|^{p-2}u(x)v(x)dx$ where $u,v\in W^{\sigma, p}(\mathbb{R}^d)$
and want to show that it is well defined.
We have some examples in my script. One time they show that for an operator B applies $|\langle Bu,v\rangle| \leq c|u|^2|v|$, where c is some constant. The other time they show that $|\langle Bu,v\rangle|< \infty$. I don't get why this shows in this two cases that the operator is well defined.
Maybe someone can explain that for me and tell me what I have to show in my case.
Edit:
$V^* = \{f:V\to\mathbb{R} |$ f linear, bounded, continuous$\}$
So we want to show that for each $u\in V$ Au is linear, bounded and continuous.
As $\langle Au, v\rangle$ is an integral it is linear and continous (right)?
So we just have to show that Au is bounded?