Theorem 27.5.2 (Taylor’s Formula, Peano Remainder).label Let $E$ be a topological vector space, $\sigma \subset \mathfrak{B}(E)$ be an ideal containing bounded subsets in finite-dimensional subspaces, $F$ be a separated locally convex space, $U \subset E$ be open, and $f: U \to F$ be $n$-fold $\tilde \sigma$-differentiable at $x_{0} \in U$, then there exists $r \in \mathcal{R}_{\sigma}^{n}(E; F)$ such that
for sufficiently small $h$.
Proof [Theorem 4.7.3, BS17]. Let
For any $1 \le k \le n$, $D^{k}_{\sigma}(x_{0}) \in B^{k}_{\sigma}(E; F)$ is symmetric by Theorem 27.4.6. Let $T_{k}(h) = \frac{1}{k!}D^{k}_{\sigma} f(x_{0})(h^{(k)})$, then by Theorem 27.4.7, for any $\bracs{t_j}_{1}^{\ell} \in E$,
so
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,
For any $A \in \sigma$ and $t > 0$,
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 $t \in (0, t_{0})$.
Since $U$ is circled,
so
and
Therefore $r \in \mathcal{R}_{\sigma}^{n+1}(E; F)$.$\square$
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