A homoegeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^n)$ is defined to be the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ under the norm \begin{equation} \lVert (-\Delta)^{s/2} f \rVert_{L^p(\mathbb{R}^n)} \end{equation} Here $s>0, p \geq 1$.
Now, I wonder if "any" $\dot{W}^{s,p}(\mathbb{R}^n)$ is a "homogeneous Banach space", in the sense that it is linearly homeomorphic to all its infinite dimensioanl subspaces. I think the name suggests it, but cannot find any relevant proof.
Also, instead of $\mathbb{R}^n$, can we think of a compact manifold like $\mathbb{T}^n:=[\mathbb{R}/\mathbb{Z}]^n$ and still get the same result?
That is, if we define $\dot{W}^{s,p}(\mathbb{T}^n)$ as the completion of the $\{ f\in C^\infty(\mathbb{T}^n) \mid \int_{\mathbb{T}^n} f=0\}$ under the above norm, is $\dot{W}^{s,p}(\mathbb{T}^n)$ still a homogeneous Banach space?
Could anyone please clarfiy for me, or provide any reference?