142 lines
4.7 KiB
TeX
142 lines
4.7 KiB
TeX
\documentclass[../../script.tex] {subfiles}
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%! TEX root = ../../script.tex
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\begin{document}
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\section{Metric and Normed spaces}
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\begin{defi}[Metric space]
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A metric space $(X, d)$ is an ordered pair consisting of a set $X$ and a mapping
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\[
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d: X \times X \longrightarrow [0, \infty]
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\]
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called metric. This mapping must fulfil the following conditions $\forall x, y, z \in X$:
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\begin{itemize}
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\item $d(x, y) \ge 0$ \text{ (Positivity)}
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\item $d(x, y) = 0 \iff x = y$ \text{ (Definedness)}
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\item $d(x, y) = d(y, x)$ \text{ (Symmetry)}
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\item $d(x, y) \le d(x, z) + d(z, y)$ \text{ (Triangle inequality)}
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\end{itemize}
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\end{defi}
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\begin{eg}
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\begin{enumerate}[(i)]
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\item Let $M$ be a set. Then
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\[
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d(x, y) = \begin{cases}
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1, & x \ne y \\
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0, & \text{else}
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\end{cases}
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\]
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is called the discrete metric.
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\item Let $X$ be the set of edges of a graph.
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\begin{align*}
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d(x, y) := &\mbox{ Minimum amount of edges that have} \\
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&\mbox{ to be passed to get from } x \mbox{ to } y
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\end{align*}
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\begin{center}
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\begin{tikzpicture}[scale=0.75]
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\node[shape=circle, draw=black, fill=black] (A) at (0, 0) {};
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\node[shape=circle, draw=black] (B) at (1, -4) {};
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\node[shape=circle, draw=black] (C) at (3, -2) {};
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\node[shape=circle, draw=black] (D) at (5, -3.5) {};
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\node[shape=circle, draw=black] (E) at (6.5, -5.5) {};
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\node[shape=circle, draw=black, fill=black] (F) at (7, -1) {};
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\path [-] (A) edge node[left] {$1$} (B);
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\path [-] (B) edge node[above left] {$2$} (C);
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\path [-] (B) edge (D);
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\path [-] (B) edge (E);
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\path [-] (C) edge (D);
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\path [-] (C) edge node[above] {$3$} (F);
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\path [-] (E) edge (F);
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\node[above=0.3cm] at (A) {$x$};
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\node[above=0.3cm] at (F) {$y$};
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\end{tikzpicture}
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\end{center}
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\item Let $X$ be the surface of a sphere.
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\[
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d(x, y) := \text{"Bee line"}
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\]
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\item Let $X$ be the set of points of the European street network.
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\[
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d(x, y) := \text{ Shortest route along this network}
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\]
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\item Let $(X, d_X)$, $(Y, d_Y)$ be metric spaces. Then
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\[
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d_{X \times Y}((x_1, y_1), (x_2, y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)
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\]
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defines a metric on $X \times Y$.
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\end{enumerate}
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\end{eg}
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\begin{defi}[Normed space]
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$(V, \dnorm)$ is said to be a normed space if $V$ is a vector space and
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\[
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\dnorm: V \longrightarrow [0, \infty)
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\]
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is a mapping (called norm) with the following properties
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\begin{itemize}
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\item $\norm{x} \ge 0$ (Positivity)
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\item $\norm{x} = 0 \iff x = 0$ (Definedness)
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\item $\norm{\lambda x} = |\lambda|\norm{x}$
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\item $\norm{x + y} \le \norm{x} + \norm{y}$ (Triangle inequality)
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\end{itemize}
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To every norm belongs a unique induced metric
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\[
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d(x, y) = \norm{x - y}
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\]
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\end{defi}
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\begin{eg}[$\realn^n$ with Euclidian norm]
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\begin{align*}
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\dnorm: \realn^n &\longrightarrow [0, \infty) \\
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(x_1, x_2, \cdots, x_n) &\longmapsto \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}
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\end{align*}
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Then $(\realn^n, \dnorm)$ is a normed space.
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\end{eg}
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\begin{eg}
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\begin{enumerate}[(i)]
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\item $(x_1, x_2, \cdots, x_n) \mapsto |x_1| + |x_2| + \cdots + |x_n|$ is also a norm on $\realn^n$.
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\item On
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\[
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V = \set[f \text{ continuous}]{f: [0, 1] \longrightarrow \realn}
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\]
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we can define the supremum norm
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\[
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\supnorm{f} = \sup \set[{x \in [0, 1]}]{|f(x)|}
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\]
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\item We can define sequence spaces as
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\[
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\ell^p = \set[\series{n} |x_n|^p < \infty]{\anyseqdef{\cmpln^n}}
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\]
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with the norm
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\[
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\norm{(x_n)}_p := \sqrt{\series{n} |x_n|^2}
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\]
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A special space is $\ell^2$, called Hilbert space
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\end{enumerate}
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\end{eg}
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\begin{rem}
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The Minkowski metric is not a metric in this sense.
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\end{rem}
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\begin{defi}[Balls and Boundedness]
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Let $\metric$ be a metric space, and $x \in X, r > 0$. We then define
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\begin{align*}
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\Oball(x) = \set[d(x, y) < r]{y \in X} && \text{Open ball} \\
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\Cball(x) = \set[d(x, y) \le r]{y \in X} && \text{Closed ball}
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\end{align*}
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A subset $M \subset X$ is called bounded if
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\[
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\exists x \in X, r > 0: ~~M \subset \Oball(x)
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\]
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\end{defi}
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\end{document} |