Various typo fixes.

This commit is contained in:
Bokuan Li
2026-03-15 23:04:46 -04:00
parent 8e7e55e8b3
commit ef91d9f91b
3 changed files with 5 additions and 5 deletions

View File

@@ -113,9 +113,9 @@
\begin{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 \autoref{proposition:chain-rule-sets-conditions} with $Dg(t) = Df((1 - t)x + ty)(y - x)$, and continuous by \autoref{proposition:derivative-sets-real}.
By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line},
By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line}, $f(y) - f(x) = g(1) - g(0)$ is contained in
\[
f(y) - f(x) = g(1) - g(0) \in \overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}} = \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}
\overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}} = \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}
\]
\end{proof}