18.5 Taylor’s Formula

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

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

$\square$

Theorem 18.5.2 (Taylor’s Formula, Peano Remainder [Theorem 4.7.3, BS17]). Let $E$ be a topological vector space, $\sigma \subset B(E)$ be an upward-directed family that includes bounded sets contained in finite-dimensional subspaces, $F$ be a separated locally convex space, $U \subset E$ be open, and $f: U \to F$ be $n$-fold $\sigma$-differentiable at $x_{0} \in U$, then there exists $r \in \mathcal{R}_{\sigma}^{n}(E; F)$ such that

\[g(x_{0} + h) = g(x_{0}) + \sum_{k = 1}^{n} \frac{1}{k!}D^{k}_{\sigma} f(x_{0})(h^{(k)}) + r(h)\]

Proof. Let

\[r(h) = g(x_{0} + h) - g(x) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k}_{\sigma} f(x_{0})(h^{(k)})\]

For any $1 \le k \le n$, $D^{k}_{\sigma}(x_{0}) \in B^{k}_{\sigma}(E; F)$ is symmetric by Theorem 18.4.3. Let $T_{k}(h) = \frac{1}{k!}D^{k}_{\sigma} f(x_{0})(h^{(k)})$, then by Proposition 18.4.4, for any $\bracs{t_j}_{1}^{\ell} \in E$,

\[D^{\ell}_{\sigma} T_{k}(h)(t_{1}, \cdots, t_{\ell}) = \begin{cases}0&\ell > k \\ D^{k}_{\sigma}(x_{0})(t_{1}, \cdots, t_{\ell})&\ell = k \\ \frac{1}{(k-\ell)!}D^{k}_{\sigma}(x_{0})(h^{(k - \ell)}, t_{1}, \cdots, t_{\ell})&\ell < k\end{cases}\]

\[D^{k}_{\sigma} r(0) = D^{k}_{\sigma} g(x_{0}) - D^{k}_{\sigma}(x_{0}) = 0\]

If $n = 1$, then the theorem holds by definition of the derivative. Now suppose inductively that the theorem holds for $n$. By the Mean Value Theorem,

\[r(h) = r(h) - r(0) \in \overline{\text{Conv}\bracs{D_\sigma r(s)(h)| s \in [0, h]}}\]

For any $A \in \sigma$ and $t > 0$,

\[\frac{r(tA)}{t^{n+1}}\subset \overline{\text{Conv}\bracs{t^{-n}D_\sigma r(s)(h)|s \in [0, h], h \in tA}}\]

Let $U \in \cn_{F}(0)$ be convex and circled, then by the inductive assumption applied to $D_{\sigma} r$, there exists $t_{0} \in (0, 1)$ such that for any $t \in (0, t_{0})$.

\[\frac{D_{\sigma} r(tA)}{t^{n}}\subset \bracs{T \in L(E; F)| T(A) \subset U}\]

Since $U$ is circled,

\[\bracs{T \in L(E; F)| T(A) \subset U}= \bracs{T \in L(E; F)| T(tA) \subset U \forall t \in (0, 1)}\]

\[\bracs{t^{-n}D_\sigma r(s)(h)|s \in [0, h], h \in tA}\subset U\]

and

\[\frac{r(tA)}{t^{n+1}}\subset \overline{\text{Conv}\bracs{t^{-n}D_\sigma r(s)(h)|s \in [0, h], h \in tA}}\subset \overline{U}\]

Therefore $r \in \mathcal{R}_{\sigma}^{n+1}(E; F)$.$\square$

Jerry's Digital Garden

18.5 Taylor’s Formula

Direct References

Jerry's Digital Garden

Direct References