Is there a name for the space of gradients?

Question

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded domain. Define the set

$$H = \left\{\nabla f : f \in \mathcal{C}^1(\Omega)\right\}.$$

I suspect that $H$ is a Hilbert space (though I am unsure about completeness) with respect to the inner product $(F,G) \in H^2 \mapsto \int_\Omega \langle F(x),G(x)\rangle \, dx$, where $\langle\cdot,\cdot\rangle$ is the standard inner product on $\mathbb{R}^d$. This inner product is well-defined because functions in $H$ are continuous and $\Omega$ is bounded. Does $H$, or variants thereof, have a name? Are there any references where its properties are studied?

Answer 1

The space in question is not complete: take a function $f$ whose derivative is in $L^2$ but is discontinuous, and consider a sequence of smooth functions converging to it in the Sobolev space $W^{1, 2}$. Then the sequence of gradients is clearly Cauchy in your space $H$ and its limit is $\nabla f \notin H$.

This tells you that the space to actually consider is $$\{\nabla f \in L^2: f \in L^1_{loc}\}$$ which is a Hilbert space. The space of $f$ with $\nabla f \in L^2$ modulo constants is denoted $\dot W^{1, 2}$ and is naturally isomorphic to the space of $L^2$ gradients. You could also identify this space of gradients with the space of exact $L^2$, $1$-forms.

I think most of these facts are in Chapter 5 -- Sobolev spaces -- of Evans' book on PDE.