Me when I forget to commit.
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Bokuan Li
@@ -1,6 +1,52 @@
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\section{Derivatives}
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\section{Derivatives and Remainders}
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\label{section:derivative}
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\begin{definition}[Derivatives and Remainders]
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\label{definition:derivative-system}
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Let $E, F$ be TVSs over $K \in \RC$ and $\ch(E; F), \calr(E; F) \subset F^E$ be vector subspaces, then $(\ch, \calr) = (\ch(E; F), \calr(E; F))$ is a pair of \textbf{derivatives} and \textbf{remainders} if
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\begin{enumerate}
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\item[(T)] For any $T \in \ch$, if there exists $V \in \cn_E(0)$ and $r \in \calr$ such that $T|_V = r|_V$, then $T = 0$.
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\end{enumerate}
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\end{definition}
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\begin{definition}[$(\ch, \calr)$-Differentiability]
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\label{definition:space-differentiability}
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Let $E, F$ be TVSs over $K \in \RC$, $(\ch, \calr)$ be a pair of derivatives and remainders, $U \subset E$ be open, $f: U \to F$ be a function, and $x_0 \in U$, then $f$ is \textbf{$(\ch, \calr)$-differentiable} at $x_0$ if there exists $V \in \cn_E(0)$, $T \in \ch$, and $r \in \calr$ such that
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\[
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f(x_0 + h) = f(x_0) + Th + r(h)
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\]
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for all $h \in V$. In which case, $T = D_{(\ch, \calr)}f(x_0)$ is the unique element of $\ch$ satisfying the above, known as the \textbf{$(\ch, \calr)$-derivative} of $f$ at $x_0$.
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\end{definition}
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\begin{proof}
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Let $S, T \in \ch$, $r, s \in \calr$, and $V \in \cn_E(0)$ such that
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\[
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f(x_0 + h) - f(x_0) = Sh + r(h) = Th + s(h)
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\]
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for all $h \in V$, then $(S - T)(h) = (s - r)(h)$. By (T), $S - T = 0$. Hence $S = T$.
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\end{proof}
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\begin{proposition}
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\label{proposition:derivative-linearity}
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Let $E, F$ be TVSs over $K \in \RC$, $(\ch, \calr)$ be a pair of derivatives and remainders, $U \subset E$ be open, $f, g: U \to F$ be functions, and $x_0 \in U$. If $f, g$ are $(\ch, \calr)$-differentiable at $x_0$, then for any $\lambda \in K$, $\lambda f + g$ is $(\ch, \calr)$-differentiable at $x_0$, and
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\[
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D_{(\ch, \calr)}(\lambda f + g)(x_0) = \lambda D_{(\ch, \calr)}f(x_0) + D_{(\ch, \calr)}g(x_0)
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\]
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\end{proposition}
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\begin{proof}
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Let $V \in \cn_E(0)$ and $r, s \in \calr$ such that
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\begin{align*}
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f(x_0 + h) - f(x_0) &= D_{(\ch, \calr)}f(x_0)h + r(h) \\
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g(x_0 + h) - f(x_0) &= D_{(\ch, \calr)}g(x_0)h + s(h)
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\end{align*}
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then
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\[
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(\lambda f + g)(x_0+h) - (\lambda f + g)(x_0) = \underbrace{[\lambda D_{(\ch, \calr)}f(x_0) + D_{(\ch, \calr)}g(x_0)]}_{\in \ch}h + \underbrace{(\lambda r + s)}_{\in \calr}(h)
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\]
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\end{proof}
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\begin{definition}[$o(t)$]
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\label{definition:little-o}
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Let $U \in \cn(0) \subset \real$ and $r: U \to \real$, then $r \in o(t)$ if
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@@ -11,13 +57,82 @@
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\begin{definition}[Tangent to $0$]
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\label{definition:tangent-to-0}
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Let $E, F$ be TVSs over $K \in \RC$, $U \in \cn_E(0)$, and $\varphi: U \to F$, then $\varphi$ is \textbf{tangent to $0$} if for any $W \in \cn_F(0)$, there exists $V \in \cn_E(0)$ circled and $r \in o(t)$ such that
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Let $E, F$ be TVSs over $K \in \RC$, $U \in \cn_E(0)$, and $\varphi: U \to F$, then $\varphi$ is \textbf{tangent to $0$} if for any $W \in \cn_F(0)$, there exists $V \in \cn_E(0)$ and $r \in o(t)$ such that
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\[
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\varphi(tV) \subset r(t)W
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\]
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for sufficiently small $t \in \real$.
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\end{definition}
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\begin{definition}[Linear Order at $0$]
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\label{definition:linear-order-at-0}
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Let $E, F$ be TVSs over $K \in \RC$, $U \in \cn_E(0)$, and $\varphi: U \to F$, then $\varphi$ is of \textbf{linear order at $0$} if for any $W \in \cn_F(0)$, there exists $V \in \cn_E(0)$ such that
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\[
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\varphi(tV) \subset tW
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\]
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for sufficiently small $t \in \real$.
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\end{definition}
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\begin{lemma}
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\label{lemma:tangent-linear-at-0}
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Let $E, F, G$ be TVSs over $K \in \RC$, $U \in \cn_E(0)$, $V \in \cn_F(0)$, $\varphi, \psi: U \to F$, and $\rho: V \to G$. If $\varphi, \psi, \rho$ are of linear order at $0$, then
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\begin{enumerate}
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\item $\rho \circ \varphi$ is of linear order at $0$.
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\item $\varphi + \psi$ is of linear order at $0$.
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\item If one of $\varphi, \rho$ is tangent to $0$, then $\rho \circ \varphi$ is tangent to $0$.
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\end{enumerate}
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If $\varphi, \psi$ are tangent to $0$, then
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\begin{enumerate}
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\item[(4)] $\varphi$ is of linear order at $0$.
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\item[(5)] $\varphi + \psi$ is tangent at $0$.
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\end{enumerate}
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Finally, suppose that $\varphi$ is linear.
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\begin{enumerate}
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\item[(6)] If $\varphi$ is continuous, then $\varphi$ is of linear order at $0$.
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\item[(7)] If $\varphi$ is tangent to $0$ and $E$ is Hausdorff, then $\varphi = 0$.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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(1): Let $W \in \cn_G(0)$, then there exists $V_0 \in \cn_F(0)$ with $\rho(tV_0) \subset tW$ and $U_0 \in \cn_E(0)$ with $\varphi(tU_0) \subset tV_0$ for sufficiently small $t$. In which case, $\rho \circ \varphi(tU_0) \subset tW$.
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(2): Let $W \in \cn_F(0)$, then there exists $W_0 \in \cn_F(0)$ with $W_0 + W_0 \subset W$ and $V_0 \in \cn_E(0)$ such that $\varphi(tV_0), \psi(tV_0) \subset tW_0$ for sufficiently small $t$. In which case,
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\[
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\varphi(tV_0) + \psi(tV_0) \subset tW_0 + tW_0 \subset tW
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\]
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(3): Let $W \in \cn_G(0)$, then there exists $V_0 \in \cn_F(0)$, $U_0 \in \cn_E(0)$, and $r \in o(t)$ such that one of the following holds for sufficiently small $t$:
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\begin{enumerate}
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\item[(a)] $\varphi(tU_0) \subset tV_0$ and $\rho(tV_0) \subset o(t)W$.
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\item[(b)] $\varphi(tU_0) \subset o(t)V_0$ and $\rho(tV_0) \subset tW$.
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\end{enumerate}
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In both cases, $\rho \circ \varphi(tU_0) \subset o(t)W$.
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(4): Let $W \in \cn_F(0)$. Using \ref{proposition:tvs-good-neighbourhood-base}, assume without loss of generality that $W$ is circled. By assumption, there exists $V_0 \in \cn_E(0)$ and $r \in o(t)$ such that
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\[
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\varphi(tV_0) \subset o(t)W = \frac{r(t)}{t} \cdot tW
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\]
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for sufficiently small $t$. Since $r \in o(t)$, $\abs{r(t)/t} \le 1$ for sufficiently small $t$. In which case, $\frac{r(t)}{t} \cdot tW \subset tW$.
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(5): Let $W \in \cn_F(0)$. Using \ref{proposition:tvs-good-neighbourhood-base}, assume without loss of generality that $W$ is circled. Let $V_0 \in \cn_E(0)$ and $r, r' \in o(t)$ such that $\varphi(tV_0) \subset r(t)W$ and $\psi(tV_0) \subset r'(t)W$ for sufficiently small $t$, then
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\[
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\varphi(tV_0) + \psi(tV_0) \subset r(t)W + r'(t)W \subset [\abs{r(t)} + \abs{r'(t)}]W
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\]
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(6): Let $W \in \cn_F(0)$, then there exists $V_0 \in \cn_E(0)$ such that $\varphi(V_0) \subset W$. In which case, $\varphi(tV_0) \subset tW$ for all $t \in \real$.
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(7): Let $x \in E$ and $W \in \cn_F(0)$. Using \ref{proposition:tvs-good-neighbourhood-base}, assume without loss of generality that $W$ is circled. Since $\varphi$ is tangent to $0$ and linear, then there exists $V \in \cn_E(0)$ and $r \in o(t)$ such that
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\[
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\varphi(tV) \subset r(t)W \quad \varphi(V) \subset \frac{r(t)}{t}W
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\]
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for sufficiently small $t \in \real$. Since $W \in \cn_F(0)$, there exists $\lambda \in K$ such that $x \in \lambda V$. Thus
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\[
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\varphi(x) \in \varphi(\lambda V) = \lambda\varphi(V) \subset \frac{\lambda r(t)}{t}W
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\]
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As $r \in o(t)$, there exists $t \in \real$ such that $\abs{\lambda r(t)/t} \le 1$, so $\varphi(x) \in \frac{\lambda r(t)}{t}W \subset W$. Since this holds for all $W \in \cn_F(0)$ and $F$ is Hausdorff, $\varphi(x) \in \ol{\bracs{0}} = \bracs{0}$ by \ref{proposition:tvs-closure} and \ref{lemma:t1}.
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\end{proof}
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\begin{lemma}
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\label{lemma:tangent-to-0}
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Let $E, F, G$ be TVSs over $K \in \RC$, $U \in \cn_E(0)$, and $\varphi: U \to F$, $\psi: U \to F$ be tangent to $0$, then:
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@@ -54,15 +169,15 @@
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As $r \in o(t)$, there exists $t \in \real$ such that $\abs{\lambda r(t)/t} \le 1$, so $\varphi(x) \in \frac{\lambda r(t)}{t}W \subset W$. Since this holds for all $W \in \cn_F(0)$ and $F$ is Hausdorff, $\varphi(x) \in \ol{\bracs{0}} = \bracs{0}$ by \ref{proposition:tvs-closure} and \ref{lemma:t1}.
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\end{proof}
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\begin{definition}[Derivative]
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\begin{definition}[(Strong Fréchet) Derivative]
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\label{definition:derivative}
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being Hausdorff, $U \subset E$ open, $f: U \to F$, and $x_0 \in U$, then $f$ is \textbf{differentiable at} $x_0$ if there exists $\lambda \in L(E; F)$, $V \in \cn_E(0)$, and $\varphi: V \to F$ tangent to $0$ such that
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being Hausdorff, $U \subset E$ open, $f: U \to F$, and $x_0 \in U$, then $f$ is \textbf{(strongly Fréchet) differentiable at} $x_0$ if there exists $\lambda \in L(E; F)$, $V \in \cn_E(0)$, and $\varphi: V \to F$ tangent to $0$ such that
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\[
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f(x_0 + h) = f(x_0) + \lambda h + \varphi(h)
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\]
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for all $h \in V$. In which case, $Df(x_0) = \lambda$ is the unique continuous linear map satisfying the above, and the \textbf{derivative} of $f$ \textbf{at} $x_0$.
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for all $h \in V$. In which case, $Df(x_0) = \lambda$ is the unique continuous linear map satisfying the above, and the \textbf{(strong Fréchet) derivative} of $f$ \textbf{at} $x_0$.
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If $f$ is differentiable at every $x_0 \in U$, then $f$ is \textbf{differentiable}, and $Df: U \to L(E; F)$ is the \textbf{derivative} of $f$.
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If $f$ is differentiable at every $x_0 \in U$, then $f$ is \textbf{(strongly Fréchet) differentiable}, and $Df: U \to L(E; F)$ is the \textbf{(strong Fréchet) derivative} of $f$.
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\end{definition}
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\begin{proof}
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Let $\lambda, \mu \in L(E; F)$ and $\varphi, \psi: V \to F$ be tangent to $0$ such that
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@@ -74,7 +189,7 @@
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\[
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0 = (\lambda - \mu)h + (\varphi - \psi)(h)
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\]
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By \ref{lemma:tangent-to-0}, $(\lambda - \mu)$ is tangent to $0$ and thus equal to $0$.
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By (7) of \ref{lemma:tangent-to-0}, $(\lambda - \mu)$ is tangent to $0$ and thus equal to $0$.
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\end{proof}
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\begin{proposition}[Chain Rule]
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By differentiability of $f$ and $g$, there exists $\varphi, \psi$ tangent to $0$ such that
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\begin{align*}
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(g \circ f)(x_0 + h) &= (g \circ f)(x_0) + Dg(f(x_0))[f(x_0 + h) - f(x_0)] + \varphi(f(x_0 + h) - f(x_0)) \\
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&= (g \circ f)(x_0) + Dg(f(x_0))[Df(x_0)(h) + \psi(h)] + \varphi(Df(x_0)(h) + \psi(h))
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&= (g \circ f)(x_0) + Dg(f(x_0))[Df(x_0)(h) + \psi(h)] + \varphi(Df(x_0)(h) + \psi(h)) \\
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&= (g \circ f)(x_0) + [Dg(f(x_0)) \circ Df(x_0)](h) \\
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&+ [Dg(f(x_0)) \circ \psi + \varphi \circ (Df(x_0) + \psi)](h)
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\end{align*}
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Since $Dg(f(x_0)) \in L(F; G)$, $Dg(f(x_0)) \circ \psi$ is tangent to $0$ by \ref{lemma:tangent-to-0}.
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On the other hand, let $W \in \cn_G(0)$, then there exists $V_1 \in \cn_F(0)$ circled and $r_1 \in o(t)$ such that
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Since $Dg(f(x_0)) \in L(F; G)$, $Dg(f(x_0)) \circ \psi$ is tangent to $0$ by (6) and (3) \ref{lemma:tangent-linear-at-0}. Similarly, $Df(x_0) + \psi$ is of linear order at $0$ by (4) and (2) of \ref{lemma:tangent-linear-at-0}, so $\varphi \circ (Df(x_0) + \psi)$ is tangent to $0$ by (3) of \ref{lemma:tangent-linear-at-0}. Thus
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\[
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\varphi(tV_1) \subset r_1(t)W
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Dg(f(x_0)) \circ \psi + \varphi \circ (Df(x_0) + \psi)
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\]
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for sufficiently small $t \in \real$. Let $V_2 \in \cn_F(0)$ such that $V_2 + V_2 \subset V_1$ and assume without loss of generality that $V_2$ is circled using \ref{proposition:tvs-good-neighbourhood-base}, then there exists $U_1 \in \cn_E(0)$ and $r_2 \in o(t)$ with
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\[
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\psi(tU_1) \subset r_2(t)V_2
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\]
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for sufficiently small $t \in \real$. In particular, since $V_2$ is circled, if $t$ is small enough, then
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\[
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\psi(tU_1) \subset r_2(t)V_2 \subset tV_2
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\]
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Thus if $U_2 = U_1 \cap Df(x_0)^{-1}(V_2)$, then
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\[
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\varphi(Df(x_0)(tU_2) + \psi(tU_2)) \subset \varphi(tV_2 + tV_2) \subset \varphi(tV_1) \subset r_1(t)W
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\]
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so $\varphi(Df(x_0)(h) + \psi(h))$ is tangent to $0$ as well.
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is tangent to $0$ by (5) of \ref{lemma:tangent-linear-at-0}.
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\end{proof}
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17
src/dg/derivative/sets.tex
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17
src/dg/derivative/sets.tex
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@@ -0,0 +1,17 @@
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\section{Differentiation With Respect to Set Systems}
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\label{section:differentiation-set-system}
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\begin{definition}[Small]
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\label{definition:differentiation-small}
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Let $E, F$ be TVSs over $K \in \RC$, $\sigma \subset B(E)$ be an upward-directed family of sets that contains all finite sets, $r: E \to F$, and $n \in \natz$, then the following are equivalent:
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\begin{enumerate}
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\item For each $A \in \sigma$, $r(th)/t^n \to 0$ uniformly on $A$.
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\item If $r_t(x) = r(tx)/t^n$, then $r_t \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^E$.
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\item For each $A \in \sigma$, $\seq{a_k} \subset A$, and $\seq{t_k} \subset K \setminus \bracs{0}$ with $t_k \to 0$ as $n \to \infty$, $r(t_ka_k)/t_k^n \to 0$ as $n \to \infty$.
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\end{enumerate}
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If the above holds, then $r$ is \textbf{$\sigma$-small of order $n$}.
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\end{definition}
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\begin{remark}
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In \ref{definition:differentiation-small}, the system $\sigma$ can be chosedn based on the bornology of $E$, and the definition of small-ness depends exclusively on $\sigma$. As such, there is an apparent disconnect between differentiation and the topology of the domain.
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\end{remark}
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