Theorem 18.5.1: Taylor's Formula, Lagrange Remainder {{\cite[Theorem 4.7.1]{Bogachev}}}

Theorem 18.5.1 (Taylor’s Formula, Lagrange Remainder [Theorem 4.7.1, BS17]). Let $-\infty < a < b < \infty$, $E$ be a separated locally convex space, $S \subset [a, b]$ be at most countable, $n \in \natp$, and $f \in C^{n}([a, b]; E)$ be $(n+1)$-fold differentiable on $[a, b] \setminus N$, then

\[f(b) - f(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k}f(a)(b - a)^{k}\]

is contained in the closed convex hullIt may be possible to sharpen the below claim to include the $1/(n+1)!$ factor. However, I was not able to follow the proof for this. of

\[\bracs{D^{n+1}f(s)(t - a)^{n+1} | s \in (a, b) \setminus N, t \in [a, b]}\]

Proof. If $n = 0$, then the theorem is the Mean Value Theorem.

Suppose inductively that the theorem holds for $n$. Let

\[g: [a, b] \to E \quad t \mapsto f(t) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(t - a)^{k}\]

then for any $t \in (0, 1)$,

\begin{align*}Dg(t)&= Df(t) - \sum_{k = 1}^{n+1}\frac{1}{(k-1)!}D^{k}f(a)(t - a)^{k-1}\\&= Df(t) - Df(a) - \sum_{k = 1}^{n}\frac{1}{k!}D^{k+1}f(a)(t - a)^{k}\end{align*}

by the Mean Value Theorem,

\[g(b) - g(a) = f(b) - f(a) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(t - a)^{k}\]

is contained in the closed convex hull of

\[\bracs{\braks{Df(t) - Df(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k+1}f(a)(t - a)^{k}}(b - a) \bigg | t \in [a, b]}\]

By the inductive hypothesis applied to $Df$, for any $t \in [a, b]$,

\[Df(t) - Df(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k+1}f(a)(t - a)^{k}\]

is contained in the closed convex hull of

\[\bracs{D^{n+2}f(s)(t - a)^{n+1} | s \in (a, t) \setminus N}\]

Therefore

\[f(b) - f(a) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(b - a)^{k}\]

is contained in the convex hull of

\[\bracs{D^{n+2}f(s)(t - a)^{n+2} | s \in (a, t) \setminus N, t \in [a, b]}\]

$\square$

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Theorem 18.3.4: Mean Value Theorem

Jerry's Digital Garden

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Jerry's Digital Garden

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