 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; otherwiseGiven $p_v\in \partial J(v)$, the Bregman distance isIt 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 distanceThe symmetric version here is the sum, not half the sum. These nonnegative quantities are not generally metrics and can vanish at distinct points.