Let $$H=H\left(\frac{\partial}{\partial x_n},\Omega \right)=\left\lbrace v\in L^2\left(\Omega\right):\frac{\partial v}{\partial x_n}\in L^2\left(\Omega\right) \right\rbrace$$ where $\Omega$ is an open subset of $\mathbb{R}^n$.
$H$ is equipped with the following inner product: $$\left(u,v\right)_H=\int_\Omega uvdx+\int_\Omega\frac{\partial u}{\partial x_n}\frac{\partial v}{\partial x_n}dx,$$ and the induced norm is denoted $\left\lVert\cdot\right\rVert_H$ or $\left\lVert\cdot\right\rVert$.
Prove that $H$ is a separable Hilbert space.
I know that for separability I should find a countable and dense subset of $H$, but I have no clue of which is this subset. Thank you for reading.
You could mimic the standard proof for $W^{1,p}$-separability. Define the map $T: H\to L^2(\Omega)^2$ by $Tu = (u, \frac\partial{\partial x_n}u)$. The clearly $T$ is an isometry. Its range is a subset of a separable space, hence separable. Since $T$ is an isometry also $H$ is separable.