Theorem 18.3.5 (Mean Value Theorem). Let $E$ be a topological vector space, $F$ be a separated locally convex space, $V \subset E$ be open and star shaped at $x \in V$, $f: V \to F$ be Gateau-differentiable on $V$, then for any $y \in V$,
\[f(y) - f(x) \in \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}\]
where $[x, y] = \bracs{(1 - t)x + ty|y \in [0, 1]}$.
Proof. Let $g: [0, 1] \to F$ be defined by $g(t) = f((1 - t)x + ty)$. Since $f$ is Gateaux-differentiable, $g$ is differentiable by the chain rule Proposition 18.2.7 with $Dg(t) = Df((1 - t)x + ty)(y - x)$, and continuous by Proposition 18.2.9.
By the Mean Value Theorem,
\[f(y) - f(x) = g(1) - g(0) \in \overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}}= \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}\]
$\square$