= Solution
For a convex functional $J:U\to\mathbb R\cup\{+\infty\}$ on a real <Banach space>, its <subdifferential> at $v$ is empty unless $J(v)$ is finite; otherwise
$$
\boxed{\partial J(v)=\{p\in U^*:J(u)\geq J(v)+\langle u-v,p\rangle\ \text{for all }u\in U\}.}
$$
Given $p_v\in\partial J(v)$, the <Bregman distance> is
$$
\boxed{D_J^{p_v}(u,v)=J(u)-J(v)-\langle u-v,p_v\rangle\geq0.}
$$
It can be infinite if $J(u)$ is infinite. If $J$ is differentiable at $v$, the choice is its derivative; otherwise the base <subgradient> must be specified. Given also $p_u\in\partial J(u)$, add both directional distances to obtain the <symmetric Bregman distance>
$$
\boxed{D_J^{\mathrm{sym}}(u,v)=D_J^{p_v}(u,v)+D_J^{p_u}(v,u)=\langle p_u-p_v,u-v\rangle.}
$$
The symmetric version here is the sum, not half the sum. These nonnegative quantities are not generally metrics and can vanish at distinct points.
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