Denote the 1-D torus as $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$. Using Fourier series, one can define $H^1(\mathbb{T})$ (see for instance this note from Wikipedia). On the other hand, the definition of $H^1(U)$ with $U=(0,1)$ is well known.
Here is my question:
How should one define the Sobolev space $H^1(\Omega)$ with $\Omega=(0,1)\times\mathbb{T}$?
[Added for clarification:] For function $f:\Omega\to\mathbb{R}$, I mean $\Omega=(0,1)\times \mathbb{T}$, not $(0,1)^2$ nor $\mathbb{T}^2$. Hence $f$ here is periodic in one component.