I am learning about Weighted Sobolev Spaces and I have been reading the book Weighted Sobolev Spaces by Alois Kufner. For $\Omega \subset \mathbb{R}^n$, the weighted Sobolev space is defined by $$W^{k,p}(\Omega, \sigma) = \{ u: \Omega \to \mathbb{R}, ||u|| < \infty \}$$ Where $$||u|| = \left( \sum \limits_{|\alpha| \leq k} \int \limits_{\Omega} |D^{\alpha} u(x) |^p \sigma (x) dx \right)^{1/p}$$
Then the author proceeds to say that if the weight $\sigma$ is bounded, then the space $W^{k,p}(\Omega, \sigma)$ is identical to the classical Sobolev space $W^{k,p}(\Omega)$.
What is the meaning of identical here? Does it mean that they have the same structure? Or maybe their norms are equivalent (they are in this case)? Or does it mean that the spaces are the exact same thing?
If they are the same, does this mean that their dual space is also the same?
In the usual Sobolev spaces, the space $W^{k,p}_0 (\Omega)$ is usually defined as the closure of $C^{\infty}_0(\Omega)$ with respect to the norm of $W^{k,p}_0 (\Omega)$.
The space $W^{k,p}_0 (\Omega, \sigma)$ can be defined similarly under some restrictions on $\sigma$ ($\sigma$ should belong to $L^1_{loc}(\Omega)$). If we again assume $\sigma$ is bounded, and by the same analogy, can we say that $W^{k,p}_0 (\Omega)$ and $W^{k,p}_0 (\Omega, \sigma)$ are identical?