From c74c37a2c5c3668fadb934be5b9709b1213d9b65 Mon Sep 17 00:00:00 2001 From: Robert Date: Sun, 28 Mar 2021 01:33:45 +0100 Subject: [PATCH] finished metric spaces --- chapters/multivar_calc.tex | 8 + chapters/sections/conv_of_funcseq.tex | 323 +++++++++++++++++++++++ chapters/sections/partial_total_diff.tex | 10 + chapters/sections/vector_spaces.tex | 2 +- chapters/topo_of_metr_spaces.tex | 1 + script.pdf | Bin 749971 -> 780104 bytes script.tex | 1 + 7 files changed, 344 insertions(+), 1 deletion(-) create mode 100644 chapters/multivar_calc.tex create mode 100644 chapters/sections/conv_of_funcseq.tex create mode 100644 chapters/sections/partial_total_diff.tex diff --git a/chapters/multivar_calc.tex b/chapters/multivar_calc.tex new file mode 100644 index 0000000..ee2c7a2 --- /dev/null +++ b/chapters/multivar_calc.tex @@ -0,0 +1,8 @@ +\documentclass[../script.tex]{subfiles} +%! TEX root = ../../script.tex + +\begin{document} + \chapter{Multivariable Calculus} + + \subfile{sections/partial_total_diff.tex} +\end{document} \ No newline at end of file diff --git a/chapters/sections/conv_of_funcseq.tex b/chapters/sections/conv_of_funcseq.tex new file mode 100644 index 0000000..e49384b --- /dev/null +++ b/chapters/sections/conv_of_funcseq.tex @@ -0,0 +1,323 @@ +\documentclass[../../script.tex]{subfiles} +% !TEX root = ../../script.tex + +\begin{document} +\section{Convergence of Function sequences} + +\begin{defi}[Pointwise convergence] + Let $M$ be a set, $f_n: M \rightarrow \field ~~\forall n \in \natn$ and $f: M \rightarrow \field$. + The sequence $\seq{f}$ is said to be pointwise convergent to $f$ if + \[ + \limn f_n(x) = f(x) ~~\forall x \in M + \] +\end{defi} + +\begin{eg} + Consider + \begin{align*} + f_n: [0, 1] &\longrightarrow \realn \\ + x &\longmapsto \begin{cases} + 1 - nx, & x \in [0, \frac{1}{n}] \\ + 0, & \text{else} + \end{cases} + \end{align*} + + \begin{center} + \begin{tikzpicture}[domain=0:1] + \begin{axis}[xmin=0, ymin=0, xmax=1, ymax=1, samples=50] + \addplot[black, thick] {1-x} node[above right, pos=0.6] {$f_1$}; + \addplot[black, thick] {1-2*x} node[above right, pos=0.4] {$f_2$}; + \addplot[black, thick] {1-3*x} node[right, pos=0.3] {$f_3$}; + \addplot[black, thick] {1-4*x} node[left, pos=0.23] {$f_4$}; + \end{axis} + \end{tikzpicture} + \end{center} + The $f_n$ are continuous for all $n \in \natn$ and converge pointwise to + \begin{align*} + f: [0, 1] &\longrightarrow \realn \\ + x &\longmapsto \begin{cases} + 1, & x = 0 \\ + 0, & x \ne 0 + \end{cases} + \end{align*} + $f$ is not continuous. +\end{eg} + +\begin{rem} + Let $M$ be a set. Then + \[ + B(M) = \set[\exists K \in \realn: ~~|f(x)| < K ~~\forall x \in M]{f_n: M \longrightarrow \field} + \] + is a linear subspace of the space of all functions $M \rightarrow \field$. We can define the supremum norm + \begin{align*} + \supnorm{\cdot}: B(M) &\longrightarrow \realn \\ + f &\longmapsto \sup_{x \in M}\set{\abs{f(x)}} + \end{align*} +\end{rem} +\begin{proof} + We will now proof that $\supnorm{\cdot}$ is a norm. + It is defined, because + \begin{equation} + \supnorm{f} = 0 \implies \abs{f(x)} = 0 ~~\forall x \in M + \end{equation} + This implies + \begin{equation} + f(x) = 0 ~~\forall x \in M \implies f = 0 + \end{equation} + The triangle inequality is proven by first considering + \begin{equation} + \abs{f(x)} \le \supnorm{f} ~~\forall f \in B(M) ~\forall x \in M + \end{equation} + Let $f, g \in B(M)$, then + \begin{equation} + \abs{f(x) + g(x)} \le \abs{f(x)} + \abs{g(x)} \le \supnorm{f} + \supnorm{g} ~~\forall x \in M + \end{equation} + Which implies + \begin{equation} + \supnorm{f + g} = \sup_{x \in M} \abs{f(x) + g(x)} \le \supnorm{f} + \supnorm{g} + \end{equation} +\end{proof} + +\begin{defi}[Uniform convergence] + A sequence of bounded functions $\seq{f}$, + \[ + f_n: M \longrightarrow \field + \] + is said to be uniformly convergent to $f: M \rightarrow \field$ if its norm converges. + \[ + \supnorm{f_n - f} \conv{n \rightarrow \infty} 0 + \] +\end{defi} + +\begin{rem} + Formally, pointwise convergence means + \[ + \forall \epsilon > 0 ~\forall x \in M ~\exists N \in \natn ~\forall n \ge N: ~~\abs{f_n(x) - f(x)} < \epsilon + \] + and uniform convergence means + \[ + \forall \epsilon > 0 ~\exists N \in \natn ~\forall x \in M ~\forall n \ge N: ~~\abs{f_n(x) - f(x)} < \epsilon + \] +\end{rem} + +\begin{thm} + The function space $B(M)$ is complete. +\end{thm} +\begin{proof} + Let $\anyseqdef[f]{B(M)}$ be a Cauchy sequence in terms of $\supnorm{\cdot}$. Firstly, we have for some fixed $x \in M$ + \begin{equation} + \abs{f_n(x) - f_m(x)} \le \supnorm{f_n - f_m} + \end{equation} + Since $\seq{f}$ is a Cauchy sequence, $(f_n(x))$ is also a Cauchy sequence in $\field_0$. Because $\field$ is complete, + $(f_n(x))$ converges, and we define + \begin{equation} + f(x) = \limn f_n(x) + \end{equation} + thus $\seq{f}$ converges pointwise to $f$. Let $\epsilon > 0$. Then + \begin{equation} + \exists N \in \natn: ~~\supnorm{f_n \cdot f_m} < \epsilon ~~\forall n, m \ge N + \end{equation} + Then $\forall x \in M, ~\forall n, m \ge N$ we have + \begin{equation} + \abs{f_n(x) - f_m(x)} \le \supnorm{f_n - f_m} < \epsilon + \end{equation} + We can find the limit for $m \rightarrow \infty$ + \begin{equation} + \abs{f(x) - f_n(x)} \le \epsilon + \end{equation} + and + \begin{equation} + \supnorm{f} = \sup_{x \in M}\abs{f} \le \sup_{x \in M} \abs{f(x) - f_n(x)} + \sup_{x \in M} \abs{f_n(x)} = \epsilon + \supnorm{f_n} + \end{equation} + Thus, $f$ is bounded. Furthermore + \begin{equation} + \supnorm{f - f_n} = \sup_{x \in M} \abs{f(x) - f_n(x)} \le \epsilon + \end{equation} + which in turn implies + \begin{equation} + \supnorm{f - f_n} \conv{n \rightarrow \infty} 0 + \end{equation} +\end{proof} + +\begin{defi} + Let $\metric$ be a metric space, then $C_b(X)$ is said to be the space of all continuous bounded functions. +\end{defi} + +\begin{rem} + If $X$ is compact (e.g. a bounded, closed subset of $\realn^n$) then all continuous functions are bounded. We then write $C(X)$ for $C_b(X)$. +\end{rem} + +\begin{thm} + Let $\metric$ be a metric space. $C_b(X)$ is closed in $B(X)$. In other words, every uniformly convergent sequence of continuous functions converges to a continuous function. +\end{thm} +\begin{proof} + Let $\anyseqdef[f]{C_b(X)}$ be a sequence that uniformly converges to $f \in B(X)$. + Let $x \in X$ and $\epsilon > 0$, then + \begin{equation} + \exists N \in \natn: ~~\supnorm{f - f_n} y \frac{\epsilon}{3} ~~\forall n \ge N + \end{equation} + Choose a fixed $n \ge N$. Since $f_n$ is continuous, this means that + \begin{equation} + \exists \delta > 0: ~~\abs{f_n(x) - f_n(y)} < \frac{\epsilon}{3} ~~\forall y \in \oball[\delta](x) + \end{equation} + Then we have for all such $y$ + \begin{equation} + \begin{split} + \abs{f(x) - f(y)} &\le \abs{f(x) - f_n(x)} + \abs{f_n(x) - f_n(y)} + \abs{f_n(y) - f(y)} \\ + &\le 2 \cdot \supnorm{f - f_n} + f_n(x) - f_n(y) < \epsilon + \end{split} + \end{equation} + This proves the continuity of $f$ in $x$. Since $x \in X$ was chosen arbitrarily, $f$ is continuous everywhere. +\end{proof} + +\begin{defi} + Let $x_0 \in \field$ and $\anyseqdef[a]{\field}$. Then + \[ + \series{n} a_n (x - x_0)^n + \] + is called a power series around $x_0$. The number + \[ + \rho := \sup\set[\series{n} a_n (x - x_0)^n \text{ converges}]{\abs{x - x_0}} + \] + is the convergence radius. + + \begin{center} + \begin{tikzpicture} + \draw[->, thick] (0, 0) -- (0, 4) node [above] {}; + \draw[->, thick] (0, 0) -- (4, 0) node [right] {}; + + \draw[fill] (2, 2) circle [radius=1pt] node [below right] {$x_0$}; + \draw[dashed] (2, 2) circle [radius=1.2cm]; + + \draw[<->, rotate around={45:(2, 2)}] (2, 2) -- node[above] {$\rho$} (3.2, 2); + + \draw[fill, rotate around={100:(2, 2)}] (3.2, 2) circle [radius=1pt] node [above] {$x$}; + \end{tikzpicture} + \end{center} +\end{defi} + +\begin{rem} + All results so far (including proofs) can be extended to $\realn^n$-valued functions, or functions with values in a Banach space in general. +\end{rem} + +\begin{thm} + Let $\series{n} a_n (x - x_0)^n$ be a power series with convergence radius $\rho \in [0, \infty) \cup \set{\infty}$. + If $\abs{x - x_0} < \rho$ then the series converges absolutely, for $\abs{x - x_0} > \rho$ it diverges. + \[ + \frac{1}{\rho} = \limsupn \sqrt[n]{\abs{a_n}} + \] +\end{thm} +\begin{proof} + W.l.o.g. choose $x_0 = 0$: + For $\abs{x} > \rho$ the series diverges by definition. If $\abs{x} < \rho$ then there exists $y \in \field$ such that + $\abs{x} < \abs{y} \le \rho$ and $\series{n} a_n y^n$ convergent. Especially, $(a_n y^n)$ is a null sequence. + This means $\exists C > 0$ such that $\abs{a_n y^n} \le C ~~\forall n \in \natn$ + \begin{equation} + \series{n} \abs{a_n x^n} = \series{n} \abs{a_n y^n} \abs{\frac{x}{y}}^n \le C \cdot \series{n} \abs{\frac{x}{y}}^n < \infty + \end{equation} + This statement only holds for $\rho > 0$. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item We have + \[ + \rho = \sup \set[\series{n}\abs{a_n}a^n \text{ converges}]{a \in [0, \infty)} + \] + + \item If the following limit exists, then + \[ + \rho = \limn \frac{\abs{a_n}}{\abs{a_{n+1}}} + \] + \end{enumerate} +\end{rem} + +\begin{eg} + The series + \[ + \series{n} x^n + \] + is convergent on $(-1, 1)$, so $\rho = 1$. The limit function is + \[ + x \longmapsto \frac{1}{1 - x} + \] +\end{eg} + +\begin{thm} + Let $\series{n} a_n (x - x_0)^n$ be a power series with convergence radius $\rho > 0$. + Let $0 < a < \rho$. Then this power series converges uniformly on $\cball[a](x_0)$. Especially + \begin{align*} + f: \oball[\rho](x_0) &\longrightarrow \realn \\ + x &\longmapsto \series{n} a_n (x_n - x_0)^n + \end{align*} +\end{thm} +\begin{proof} + W.l.o.g. choose $x_0 = 0$. Let $0 < a < \rho$. We know that $\series{n} a_nx^n$ converges on $\cball[a](0)$. + + \begin{center} + \begin{tikzpicture} + \draw[->, thick] (0, -3) -- (0, 3); + \draw[->, thick] (-3, 0) -- (3, 0); + + \draw[fill] (0, 0) circle [radius=1pt]; + + \draw[solid] (0, 0) circle [radius=1.8cm]; + \draw[dashed] (0, 0) circle [radius= 2.4cm] node[above right=2.4cm] {$\rho$}; + + \node at (2, 0.2) {$a$}; + \end{tikzpicture} + \end{center} + + Define + \begin{equation} + \begin{split} + f_n: \cball[a](0) &\longrightarrow \field \\ + x &\longmapsto x^n ~~\forall n \in \natn + \end{split} + \end{equation} + We can see that + \begin{equation} + \supnorm{f} = \sup_{x \in \cball[a](0)} \abs{f_n} = \sup_{x \in \cball[a](0)} = a^n + \end{equation} + and thus + \begin{equation} + \series{n} a_n f_n \implies \series{n} \supnorm{a_n f_n} = \series{n} \abs{a_n}^n < \infty + \end{equation} + because $a < \rho$. The series $\series{n} a_n f_n$ is absolutely convergent in $C(\cball[a](0))$. + Since $C(\cball[a](0))$ is complete, $\series{n} a_n f_n$ is convergent because the partial sums $\series[N]{n} a_n f_n$ are continuous $\forall N \in \natn$. + Therefore $f$ is also continuous on $\cball[a](0)$. Let $x \in \oball[\rho](0)$. Then there exists some $a > 0$ such that $\abs{x} < a < \rho$. + Thus, $f$ is continuous on $\cball[a](0)$. Since $\cball[a](0)$ contains a neighbourhood of $x$, and continuity is a local property, + $f$ is also continuous in $x$. Because $x \in \oball[\rho](0)$ was chosen arbitrarily, $f$ is continuous. +\end{proof} + +\begin{rem} + $\exp$, $\sin$, $\cos$ are continuous. +\end{rem} + +\begin{eg} + The statements above can be extended to Banach space-valued power series (e.g. matrix-valued functions). + The norm on $\realn^{n \times n}$ is + \[ + \norm{A} = \sup\set[{\forall x \in \oball[1](0)}]{\norm{Ax}} + \] + Define + \[ + \exp(A) := \series{0} \frac{A^n}{n!} + \] + This converges $\forall A \in \realn^{n \times n}$, because + \begin{align*} + \series{n} \norm{\frac{A^n}{n!}} &= \series{n} \frac{1}{n!} \norm{A^n} \le \series{n} \frac{1}{n!} \norm{A}^n \\ + &= \exp(\norm{A}) < \infty + \end{align*} + Thus, $\series{n} \frac{A^n}{n!}$ converges absolutely. Now consider the function + \begin{align*} + \realn &\longrightarrow \realn^{n \times n} \\ + t &\longmapsto \exp(At) + \end{align*} + This is a matrix-valued power series + \[ + \exp(At) = \series{n} \frac{(At)^n}{n!} = \series{n} \frac{A^n}{n!} t^n + \] + with a convergence radius of $\rho = \infty$. In this case $\exp(A + B)$ doesn't necessarily have to equal $\exp(A) \cdot \exp(B)$. +\end{eg} +\end{document} \ No newline at end of file diff --git a/chapters/sections/partial_total_diff.tex b/chapters/sections/partial_total_diff.tex new file mode 100644 index 0000000..e91ae6d --- /dev/null +++ b/chapters/sections/partial_total_diff.tex @@ -0,0 +1,10 @@ +\documentclass[../../script.tex]{subfiles} +%! TEX root = ../../script.tex + +\begin{document} +\section{Partial and Total Differentiability} + +\begin{defi} + +\end{defi} +\end{document} \ No newline at end of file diff --git a/chapters/sections/vector_spaces.tex b/chapters/sections/vector_spaces.tex index 989d2c6..b868d47 100644 --- a/chapters/sections/vector_spaces.tex +++ b/chapters/sections/vector_spaces.tex @@ -304,7 +304,7 @@ Let $V$ be a vector space. A non-empty set $W \subset V$ is called a vector subs \begin{eg} Consider \[ - W = \set[chi \in \realn]{(\chi, \chi) \in \realn^2} + W = \set[\chi \in \realn]{(\chi, \chi) \in \realn^2} \] This is a subspace, because \[ diff --git a/chapters/topo_of_metr_spaces.tex b/chapters/topo_of_metr_spaces.tex index c92cc63..d0fc8c0 100644 --- a/chapters/topo_of_metr_spaces.tex +++ b/chapters/topo_of_metr_spaces.tex @@ -8,4 +8,5 @@ \subfile{sections/seq_ser_limits.tex} \subfile{sections/open_closed_sets.tex} \subfile{sections/continuity.tex} + \subfile{sections/conv_of_funcseq.tex} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 4f5d3daacfac80c4677b861c1171be41097a46b6..7de147747f94d092d19e48980d5d09092b076aca 100644 GIT binary patch delta 174603 zcmb?^cR&TV6_8)lqWI@j=b@spRRsPQKik%5aLMD*uJ4c3@6+_*`gDWtN!7WVLc0x(PW!BQr~S7}y_{FP z()h!xX6?HE^l|ec>EAY6*rCXmf5eX{z2j-cmR)y6k(nJe2GaMG7-=+_O!RwwN;mA> zzxTj?9Xt1rFquf?TWxGvQ;OVDqhIfVeNdY$lc~LW^&ZeaqE&<`(axYUB__%Lo0R_( 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