Consider the gradient (in the weak sense) as an operator $\nabla \colon H^1(\Omega)/\mathbb{R} \to [L^2(\Omega)]^d$, where $\Omega \subset \mathbb R^d$ is a domain with a smooth boundary and $d\in\{2,3\}$ is the space dimension.
Here, $[L^2(\Omega)]^d$ denotes the space of tuples or triples of square integrable functions and $H^1(\Omega)$ is the space of $L^2(\Omega)$ functions with weak derivatives in $L^2(\Omega)$. Then $H^1(\Omega)/\mathbb R$ is made up by the equivalence classes of functions that differ only by a constant.
Is this $\nabla$ an homeomorphism on its range, i.e. is it injective and is its image closed in $[L^2(\Omega)]^d$?
EDIT thanks to @5pm for the hints
Summarizing the hints by @5pm, we can state:
Since the norm in $H^1(\Omega)/\mathbb{R}$ is equivalent to the $L^2$-norm of the gradient, one has that the $\nabla$ as defined above is bounded from below. Thus, $\nabla$ is injective and, as it is linear, also closed.