From 538a02ba37448786b3b6813b224f64684aaa90b8 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 10 May 2026 19:42:25 -0400 Subject: [PATCH] Added the inverse function theorem. --- document.tex | 1 + src/dg/derivative/index.tex | 1 + src/dg/derivative/inverse.tex | 52 ++++++++++++++++++++++++++++++++++ src/dg/index.tex | 2 +- src/op/banach/definitions.tex | 18 ++++++++++++ src/op/banach/index.tex | 5 ++++ src/op/banach/invertible.tex | 52 ++++++++++++++++++++++++++++++++++ src/op/index.tex | 5 ++++ src/op/notation.tex | 8 ++++++ src/topology/metric/metric.tex | 30 ++++++++++++++++++++ 10 files changed, 173 insertions(+), 1 deletion(-) create mode 100644 src/dg/derivative/inverse.tex create mode 100644 src/op/banach/definitions.tex create mode 100644 src/op/banach/index.tex create mode 100644 src/op/banach/invertible.tex create mode 100644 src/op/index.tex create mode 100644 src/op/notation.tex diff --git a/document.tex b/document.tex index 738beba..cf0cbc6 100644 --- a/document.tex +++ b/document.tex @@ -11,6 +11,7 @@ Hello this is all my notes. \input{./src/fa/index} \input{./src/measure/index} \input{./src/dg/index} +\input{./src/op/index} %\input{./src/process/index} \bibliographystyle{alpha} % We choose the "plain" reference style diff --git a/src/dg/derivative/index.tex b/src/dg/derivative/index.tex index b6cf66c..d4ebb80 100644 --- a/src/dg/derivative/index.tex +++ b/src/dg/derivative/index.tex @@ -7,3 +7,4 @@ \input{./higher.tex} \input{./taylor.tex} \input{./power.tex} +\input{./inverse.tex} diff --git a/src/dg/derivative/inverse.tex b/src/dg/derivative/inverse.tex new file mode 100644 index 0000000..c5b6939 --- /dev/null +++ b/src/dg/derivative/inverse.tex @@ -0,0 +1,52 @@ +\section{Inverse Mappings} +\label{section:inverse-function-theorem} + +\begin{theorem}[Inverse Function Theorem] +\label{theorem:inverse-function-theorem} + Let $E$ be a Banach space, $U \subset E$ be open, $p \ge 1$, $f \in C^p(U; E)$ be $p$-times continuously Fréchet-differentiable, and $x_0 \in U$. If $Df(x_0)$ is an isomorphism, then: + \begin{enumerate} + \item There exists $V \in \cn_E(x_0)$ such that $f|_V$ is a $C^p$-isomorphism. + \item Let $f^{-1}: f(V) \to V$ be the local inverse of $f$ on $V$, then $Df^{-1}(x_0) = [Df(x_0)]^{-1}$. + \end{enumerate} + +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem XIV.1.2]{Lang}}}. ] + By translation, assume without loss of generality that $x_0 = f(x_0) = 0$ and $Df(x_0) = Df(0) = I$. + + \textit{Existence and Uniqueness of Inverse}: Since $f \in C^1$, there exists $r > 0$ such that $\norm{Df(x) - I}_{L(E; E)} < 1/2$ for all $x \in \ol{B_E(0, r)}$. In which case, by \autoref{lemma:neumann-series}, $Df(x)$ is an isomorphism for all $x \in B(0, r)$. Let + \[ + g: \overline{B_E(0, r)} \to E \quad x \mapsto x - f(x) + \] + + For any $x, y \in \overline{B_E(0, r)}$, by the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem}, + \[ + \norm{g(x) - g(y)}_E \le \norm{x}_E \cdot \sup_{y \in \overline{B_E(0, r)}}\norm{Dg(y)}_E \le \frac{\norm{x - y}_E}{2} + \] + + In particular, for any $x \in \overline{B_E(0, r)}$, $\norm{g(x)}_E = \norm{g(x) - g(0)}_E \le \norm{x}_E/2$, so $g: \ol{B_E(0, r)} \to \ol{B_E(0, r/2)}$ is a contraction. + + For each $y \in B(0, r/2)$, the mapping + \[ + g_y: \overline{B(0, r)} \to \overline{B(0, r)} \quad x \mapsto x - f(x) + y + \] + + is also a contraction. By \hyperref[Banach's Fixed Point Theorem]{theorem:banach-fixed-point}, there exists a unique $x \in B(0, r)$ such that $g_y(x) = x$. In which case, $f(x) = y$. Therefore $f$ restricted to $V = f^{-1}(B(0, r))$ is invertible. + + \textit{Differentiability of Inverse}: Let $f^{-1}: f(V) \to V$ be the local inverse of $f$ on $V$. By assumption, it is sufficient to show that $Df^{-1}(0) = I$ as well. For each $y \in \overline{B(0, r/2)}$, + \begin{align*} + \norm{f^{-1}(y) - y}_E &= \norm{f^{-1}(y) - f(f^{-1}(y))}_E \\ + &= \norm{f^{-1}(y) - f^{-1}(y) - r(f^{-1}(y))}_E = \norm{r(f^{-1}(y))}_E + \end{align*} + + where $r(x)/\norm{x}_E \to 0$ as $x \to 0$. In addition, + \begin{align*} + \norm{f^{-1}(y)}_E &= \norm{f^{-1}(y) - y + y}_E \\ + &\le \norm{g(f^{-1}(y))}_E + \norm{y}_E \le 2\norm{y}_E + \end{align*} + + so $[f^{-1}(y) - y]/\norm{y}_E \to 0$ as $y \to 0$. Therefore $f^{-1}$ is differentiable at $0$ with $Df^{-1} = I$. + + \textit{Smoothness of Inverse}: By the above argument, the inverse is differentiable on every point in $B(0, r/2)$, and $Df^{-1}(f(x)) = [Df(x)]^{-1}$ for all $x \in V$. By \autoref{proposition:banach-algebra-inverse}, the inversion map $T \mapsto T^{-1}$ is smooth. Therefore if $Df \in C^{p - 1}$, then $f \in C^{p - 1}$ as well. +\end{proof} + + diff --git a/src/dg/index.tex b/src/dg/index.tex index 0e5023f..5f53eb1 100644 --- a/src/dg/index.tex +++ b/src/dg/index.tex @@ -1,4 +1,4 @@ -\part{Differential Geometry} +\part{Calculus} \label{part:diffgeo} \input{./derivative/index.tex} diff --git a/src/op/banach/definitions.tex b/src/op/banach/definitions.tex new file mode 100644 index 0000000..3ef240d --- /dev/null +++ b/src/op/banach/definitions.tex @@ -0,0 +1,18 @@ +\section{Banach Algebras} +\label{section:banach-algebras} + +\begin{definition}[Banach Algebra] +\label{definition:banach-algebra} + Let $A$ be an associative algebra over $\complex$ and $\norm{\cdot}_A: A \to [0, \infty)$ be a norm, then $A$ is a \textbf{Banach algebra} if: + \begin{enumerate} + \item $A$ is complete with respect to $\norm{\cdot}_A$. + \item For any $x, y \in A$, $\norm{xy}_A \le \norm{x}_A\norm{y}_A$. + \end{enumerate} +\end{definition} + +\begin{definition}[Unital Banach Algebra] +\label{definition:unital-banach-algebra} + Let $A$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, $1$ is the unique \textbf{multiplicative identity} of $A$. +\end{definition} + + diff --git a/src/op/banach/index.tex b/src/op/banach/index.tex new file mode 100644 index 0000000..8187989 --- /dev/null +++ b/src/op/banach/index.tex @@ -0,0 +1,5 @@ +\chapter{$C*$-Algebras} +\label{chap:banach-algebras} + +\input{./definitions.tex} +\input{./invertible.tex} \ No newline at end of file diff --git a/src/op/banach/invertible.tex b/src/op/banach/invertible.tex new file mode 100644 index 0000000..2798e99 --- /dev/null +++ b/src/op/banach/invertible.tex @@ -0,0 +1,52 @@ +\section{Invertible Elements} +\label{section:invertible-elements} + + +\begin{definition}[Invertible] +\label{definition:banach-algebra-invertible} + Let $A$ be a unital Banach algebra and $x \in A$, then $x$ is \textbf{invertible} if there exists $x^{-1} \in A$ such that $xx^{-1} = x^{-1}x = 1$. The set $G(A)$ denotes the collection of all invertible elements in $A$. +\end{definition} + +\begin{lemma} +\label{lemma:neumann-series} + Let $A$ be a unital banach algebra and $x \in B_A(1, 1)$, then $x \in G(A)$ with + \[ + x^{-1} = \sum_{n = 0}^\infty (1 - x)^n + \] +\end{lemma} +\begin{proof} + Since $\norm{1 - x}_A < 1$, the series converges absolutely. Let $y = \sum_{n = 0}^\infty (1 - x)^n$, then + \[ + (1 - x) \sum_{n = 0}^\infty (1 - x)^n = \sum_{n = 0}^\infty (1 - x)^n - 1 = \sum_{n = 0}^\infty (1 - x)^n (1 - x) + \] + + so $(1 - x)y = y - 1 = y(1 - x)$, and $xy = yx = 1$. +\end{proof} + +\begin{proposition} +\label{proposition:banach-algebra-inverse} + Let $A$ be a unital Banach algebra, then: + \begin{enumerate} + \item $G(A)$ is open. + \item For any $x \in G(A)$ and $y \in B_A(0, \normn{x^{-1}}_A^{-1})$, + \[ + (x - y)^{-1} = x^{-1}\sum_{n = 0}^\infty (yx^{-1})^n + \] + + \item The map $G(A) \to G(A)$ defined by $x \mapsto x^{-1}$ is $C^\infty$. + \end{enumerate} +\end{proposition} +\begin{proof} + (2): For any $x \in G(A)$ and $y \in B(0, \normn{x^{-1}}_A^{-1})$, $(x - y) = (1 - yx^{-1})x$. By \autoref{lemma:neumann-series}, + \[ + (1 - yx^{-1})^{-1} = \sum_{n = 0}^\infty (yx^{-1})^n + \] + + so + \[ + (x - y)^{-1} = x^{-1}\sum_{n = 0}^\infty (yx^{-1})^n + \] + + (3): Since the inversion map is locally a power series, it is $C^\infty$ by \autoref{theorem:termwise-differentiation}. +\end{proof} + diff --git a/src/op/index.tex b/src/op/index.tex new file mode 100644 index 0000000..6874a17 --- /dev/null +++ b/src/op/index.tex @@ -0,0 +1,5 @@ +\part{Operator Algebras} +\label{part:operator-algebras} + +\input{./banach/index.tex} +\input{./notation.tex} \ No newline at end of file diff --git a/src/op/notation.tex b/src/op/notation.tex new file mode 100644 index 0000000..c57bc82 --- /dev/null +++ b/src/op/notation.tex @@ -0,0 +1,8 @@ +\chapter{Notations} +\label{chap:op-notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + $1$ & Identity element of a unital algebra. & \autoref{definition:unital-banach-algebra} \\ +\end{tabular} \ No newline at end of file diff --git a/src/topology/metric/metric.tex b/src/topology/metric/metric.tex index ec4c0de..a1d7709 100644 --- a/src/topology/metric/metric.tex +++ b/src/topology/metric/metric.tex @@ -41,3 +41,33 @@ (3) $\Rightarrow$ (1): Let $\seq{x_n} \subset X$ be a countable dense subset. Let $x \in X$ and $k \in \natp$, then there exists $x_n \in \natp$ such that $d(x, x_n) < 1/(2k)$. In which case, $x \in B(x_n, 1/(2k)) \subset B(x_n, 1/k)$. Therefore $\bracs{B(x_n, 1/k)|n, k \in \natp}$ forms a countable basis for $X$. \end{proof} + +\begin{theorem}[Banach's Fixed Point Theorem] +\label{theorem:banach-fixed-point} + Let $(X, d)$ be a metric space and $f: X \to X$. If there exists $C \in (0, 1)$ such that + \[ + d(f(x), f(y)) \le Cd(x, y) \quad \forall x, y \in X + \] + + then: + \begin{enumerate} + \item There exists a unique $x \in X$ such that $f(x) = x$. + \item For any $y \in X$, $\limv{n}f^n(y) = x$. + \end{enumerate} +\end{theorem} +\begin{proof} + Let $x_0 \in X$ be arbitrary, and $x_n = f^n(x_0)$, then for ecah $n \in \natp$, + \[ + d(x_n, x_{n+1}) \le C d(x_{n-1}, x_n) \le C^n d(x_0, x_1) + \] + + Thus $\seq{x_n} \subset X$ is Cauchy, and converges to a point $x \in X$. + + (2): For any $y_0 \in X$, let $y_n = f^n(y_0)$, then $d(x_n, y_n) \to 0$ as $n \to \infty$, so $\limv{n}f^n(y_0) = x$. + + (1): Since $f$ is Lipschitz continuous, + \[ + f(x) = f\braks{\limv{n}f^n(x)} = \limv{n}f^{n+1}(x) = x + \] +\end{proof} +