From 2e00ac6f10bafdde091600b12c900c2f7db6c88c Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 8 May 2026 18:51:09 -0400 Subject: [PATCH] Adjusted the interchange of limits and derivaties. --- src/dg/derivative/higher.tex | 2 +- src/dg/derivative/sets.tex | 38 ++++++++++++++++++------------------ 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/src/dg/derivative/higher.tex b/src/dg/derivative/higher.tex index b5c679a..6e49104 100644 --- a/src/dg/derivative/higher.tex +++ b/src/dg/derivative/higher.tex @@ -47,7 +47,7 @@ \begin{definition}[Space of Differentiable Functions] \label{definition:differentiable-space} - Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$, then $D_\sigma^k(U; F)$ is the \textbf{space of $n$-fold $\sigma$-differentiable functions} from $U$ to $F$. + Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$, then $D_\sigma^k(U; F)$/$\tilde D_\sigma^k(U; F)$ is the \textbf{space of $n$-fold $\sigma$/$\tilde \sigma$-differentiable functions} from $U$ to $F$. \end{definition} \begin{theorem}[Symmetry of Higher Derivatives] diff --git a/src/dg/derivative/sets.tex b/src/dg/derivative/sets.tex index cdc54b2..0a05d80 100644 --- a/src/dg/derivative/sets.tex +++ b/src/dg/derivative/sets.tex @@ -130,24 +130,24 @@ \begin{theorem}[Interchange of Limits and Derivatives] \label{theorem:differentiable-uniform-limit} - Let $E$ be a TVS over $K \in \RC$, $F$ be a separated locally convex space over $K$, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$. Let $\fF \subset 2^{D_\sigma^n(U; F)}$ be a filter such that: + Let $E$ be a TVS over $K \in \RC$, $F$ be a separated locally convex space over $K$, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$. Let $\fF \subset 2^{\tilde D_\sigma^n(U; F)}$ be a filter such that: \begin{enumerate}[label=(\alph*)] \item There exists $f: U \to F$ such that $\fF \to f$ pointwise. - \item For each $1 \le k \le n$, there exists $f^{(k)}: U \to L^{(k)}_\sigma(E; F)$ such that for all $x \in U$, and $A \in \sigma$ with $x + [0, 1]A \subset U$, $D_\sigma^k(\fF) \to f^{(k)}$ uniformly on $x + [0, 1]A$. + \item For each $1 \le k \le n$, there exists $f^{(k)}: U \to B^{k}_\sigma(E; F)$ such that for all $x \in U$, and $A \in \sigma$ with $x + [0, 1]A \subset U$, $D_\sigma^k(\fF) \to f^{(k)}$ uniformly on $x + [0, 1]A$. \end{enumerate} - then $f \in D_\sigma^n(U; F)$ and $D^k_\sigma f = f^{(k)}$ for all $1 \le k \le n$. In particular, if $\sigma$ is saturated, then $(b)$ may be replaced by + then $f \in \tilde D_\sigma^n(U; F)$ and $D^k_\sigma f = f^{(k)}$ for all $1 \le k \le n$. In particular, if $\sigma$ is saturated, then $(b)$ may be replaced by \begin{enumerate} - \item[(b)] For each $1 \le k \le n$, there exists $f^{(k)}: U \to L^{(k)}_\sigma(E; F)$ such that $D_\sigma^k(\fF) \to f^{(k)}$ uniformly on every $A \in \sigma$. + \item[(b)] For each $1 \le k \le n$, there exists $f^{(k)}: U \to B^{k}_\sigma(E; F)$ such that $D_\sigma^k(\fF) \to f^{(k)}$ uniformly on every $A \in \sigma$. \end{enumerate} \end{theorem} \begin{proof} - Assume without loss of generality that $n = 1$. For any $\varphi \in D^1_\sigma(U; F)$, $x \in U$, and $h \in E$ such that $x + h \in U$, + Assume without loss of generality that $n = 1$. For any $\varphi \in \tilde D^1_\sigma(U; F)$, $x \in U$, and $h \in E$ such that $x + h \in U$, \begin{align*} - f(x + h) - f(x) - f^{(1)}(x)h &= \underbrace{\varphi(x + h) - \varphi(x) - D\varphi(x)h}_{\in \mathcal{R}_\sigma(E; F)} \\ + f(x + h) - f(x) - f^{(1)}(x)h &= \underbrace{\varphi(x + h) - \varphi(x) - D_\sigma\varphi(x)h}_{\in \mathcal{R}_\sigma(E; F)} \\ &+ (f - \varphi)(x + h) - (f - \varphi)(x) \\ - &+ (D\varphi - f^{(1)})(x)h + &+ (D_\sigma\varphi - f^{(1)})(x)h \end{align*} Since $\fF \to f$ pointwise, for any $S \in \fF$, @@ -155,45 +155,45 @@ (f - \varphi)(x + h) - (f - \varphi)(x) \in \overline{\bracs{(g - \varphi)(x + h) - (g - \varphi)(x)|g \in S}} \] - By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem}, for any $g \in D^1_\sigma(U; F)$, + By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem}, for any $g \in \tilde D^1_\sigma(U; F)$, \[ - (g - \varphi)(x + h) - (g - \varphi)(x) \in \ol{\conv}\bracs{D(g - \varphi)(x + th)h|t \in [0, 1]} + (g - \varphi)(x + h) - (g - \varphi)(x) \in \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + th)h|t \in [0, 1]} \] Hence \[ - (f - \varphi)(x + h) - (f - \varphi)(x) \in \ol{\conv}\bracs{D(g - \varphi)(x + th)h|g \in S, t \in [0, 1]} + (f - \varphi)(x + h) - (f - \varphi)(x) \in \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + th)h|g \in S, t \in [0, 1]} \] so for any $t \in (0, 1)$ and $A \in \sigma$, \begin{align*} &(f - \varphi)(x + tA) - (f - \varphi)(x) \\ - &\subset \ol{\conv}\bracs{D(g - \varphi)(x + sth)th|g \in S, s \in [0, 1], h \in A} \\ - &= t\ol{\conv}\bracs{D(g - \varphi)(x + sth)h|g \in S, s \in [0, 1], h \in A} + &\subset \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sth)th|g \in S, s \in [0, 1], h \in A} \\ + &= t\ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sth)h|g \in S, s \in [0, 1], h \in A} \end{align*} and \begin{align*} &t^{-1}[(f - \varphi)(x + tA) - (f - \varphi)(x)] \\ - &\subset \ol{\conv}\bracs{D(g - \varphi)(x + sth)h|g \in S, s \in [0, 1], h \in A} \\ - &\subset \ol{\conv}\bracs{D(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} + &\subset \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sth)h|g \in S, s \in [0, 1], h \in A} \\ + &\subset \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} \end{align*} - In addition, since $D(\fF) \to f^{(1)}$ pointwise, + In addition, since $D_\sigma(\fF) \to f^{(1)}$ pointwise, \[ - t^{-1}(f^{(1)} - D\varphi)(x)(tA) \subset \ol{\conv}\bracs{D(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} + t^{-1}(f^{(1)} - D_\sigma\varphi)(x)(tA) \subset \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} \] as well. Now, let $V \in \cn_F(0)$ be convex and circled. Using assumption (b), let $S \in \fF$ such that for any $\varphi \in S$, \[ - \ol{\conv}\bracs{D(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} \subset V + \ol{\conv}\bracs{D_\sigma(g - \varphi)(x + sh)h|g \in S, s \in [0, 1], h \in A} \subset V \] Fix $\varphi \in S$, then as $\varphi$ is differentiable at $x$, there exists $\delta \in (0, 1)$ such that \[ - t^{-1}[\varphi(x + tA) - \varphi(x) - D\varphi(x)(tA)] \subset V + t^{-1}[\varphi(x + tA) - \varphi(x) - D_\sigma\varphi(x)(tA)] \subset V \] for all $t \in (0, \delta)$. @@ -203,7 +203,7 @@ t^{-1}[f(x + tA) - f(x) - f^{(1)}(x)(tA)] \subset 3V \] - for all $t \in (0, \delta)$. Therefore $f$ is $\sigma$-differentiable at $x$ with $D_\sigma f(x) = f^{(1)}(x)$. + for all $t \in (0, \delta)$. Therefore $f$ is $\tilde \sigma$-differentiable at $x$ with $D_\sigma f(x) = f^{(1)}(x)$. \end{proof}