I have been trying to understand the Gateaux differential (if it does exist) of the total variation norm ($\|\cdot\|_\text{TV}$) over the space of measure $M(\textit{X}, \mathbb{R}^k)$ where $\textit{X}$ is a compact subset of some $\mathbb{R}^n$ space. Does it exist under some topology over the space?
The closet I could come up was this result on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$:
$\textbf{Proposition (Subdifferential of the total variation)}$ Let us endow $M(\mathbb{T})$ with the weak-* topology and $C(\mathbb{T})$ with the weak topology. Then, for any $m \in M(\mathbb{T})$, we have: $$ \partial \|m\|_\text{TV} = \left\{ \eta \in C(\mathbb{T}) \, ; \, \|\eta\|_{\infty} \leq 1 \text{ and } \int \eta \, dm = \|m\|_\text{TV} \right\}.$$
where $C(\mathbb{T})$ is the space of continuous functions that vanish at the infinity.
I'm unable to interpret that set. Is there an alternate way to write this? Does it have certain properties that could be used to distinguish the sets corresponding to two different measures $m$ and $m'$?
