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Adjusted organisation in the TVS chapter.

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Bokuan Li
2026-05-05 21:58:54 -04:00
parent 97372173e1
commit ce56f5d167
12 changed files with 267 additions and 253 deletions
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src/dg/derivative/taylor.tex
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@@ -62,7 +62,7 @@
\begin{theorem}[Taylor's Formula, Peano Remainder]
\label{theorem:taylor-peano}
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
Let $E$ be a topological vector space, $\sigma \subset \mathfrak{B}(E)$ 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
\[
g(x_0 + h) = g(x_0) + \sum_{k = 1}^n \frac{1}{k!}D^k_\sigma f(x_0)(h^{(k)}) + r(h)
\]