Proposition 18.2.2.label Let $E$ be a locally convex space over $\real$, $f, g: E \to (-\infty, \infty]$, and $x \in \bracs{f, g < \infty}$, then:

1. (1)

   $\partial f(x)$ is a weak*-closed convex set.

2. (2)

   $\partial(f + g)(x) \supset \partial f(x) + \partial g(x)$.