Definition 18.2.1 (Subdifferential).label Let $E$ be a locally convex space over $\real$, $f: E \to (-\infty, \infty]$, $x \in \bracs{f < \infty}$, and $\phi \in E^{*}$, then $\phi$ is a subgradient of $f$ if for any $h \in E$,
\[f(x + h) \ge f(x) + \dpn{h, \phi}{E}\]
The set $\partial f(x)$ of all subgradients of $f$ at $x$ is the subdifferential of $f$ at $x$, and the mapping $\partial f$ is the subdifferential of $f$.
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