Adjusted the interchange of limits and derivaties.

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]

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} \\
+    &\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(g - \varphi)(x + sth)h|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_\sigma(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 + 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}