02. Metric Spaces

2.1 The real numbers and the axiom of choice

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2.2 The definition of a metric space

Distance Function

Let $X$ be a set

A distance function on $X$ is a function $d:X\times X\to \mathbb{R}$ with (for all $x,y,z\in X$)

1. Positivity:

$$d(x,y)\ge 0\text{ and }d(x,y)=0⟺x=y$$

2. Symmetry:

$$d(x,y)=d(y,x)$$

3. Triangle Inequality:

$$d(x,z)\le d(x,y)+d(y,z)$$

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Metric Space

Pair $(X,d)$ consisting of a set $X$ with distance function $d$

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Reverse Triangle Inequality of Metric Spaces lemma

Let $x,y,z$ be points of a metric space then

$$|d(x,y)-d(x,z)|\le d(y,z)$$

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2.3 Some examples of metric spaces

Examples of Distance Functions

Take $X={\mathbb{R}}^n$ then define the following on $X$

$$\begin{array}{rl}{d}_1(v,w)& =\sum _{i=1}^n|v_i-w_i|\\ d_2(v,w)& ={\left(\sum _{i=1}^n(v_i-w_i{)}^2\right)}^{1/2}\\ d_{\infty }(v,w)& =\underset{i\in \{1,2,\cdots ,n\}}{\max }|v_i-w_i|\end{array}$$

Relation of $d_2$ to Euclidean Norm Euclidean Norm $||v|{|}_2$ of a vector $v=(v_1,\cdots ,v_n)\in {\mathbb{R}}^n$ is

$$||v|{|}_2:={\left(\sum _{i=1}^n|v_i{|}^2\right)}^{1/2}=⟨v,v{⟩}^{1/2}$$

where the inner product is defined by

$$⟨v,w⟩=\sum _{i=1}^n{v}_i{w}_i$$

Hence $d_2(v,w)=||v-w|{|}_2$ so the triangle inequality is

$$||u-w|{|}_2\le ||u-v|{|}_2+||v-w|{|}_2$$

Property of Euclidean Norm lemma If $x,y\in {\mathbb{R}}^n$ then

$$||x+y|{|}_2\le ||x|{|}_2+||y|{|}_2$$

Proof

Since $||v|{|}_2\ge 0$ for all $v\in {\mathbb{R}}^n$ then the inequality is equivalent to

$$||x+y|{|}_2^2\le ||x|{|}_2^2+2||x|{|}_2||y|{|}_2+||y|{|}_2^2$$

As $||x+y|{|}_2^2=⟨x+y,x+y⟩=||x|{|}_2^2+2⟨x,y⟩+||y|{|}_2^2$
Then by Cauchy-Schawrz Inequality

$$|⟨x,y⟩|\le ||x|{|}_2||y|{|}_2$$

Hence the result follows by the above inequality

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2.4 Norms

Norms

Let $V$ be any vector space (over $\mathbb{R}$)

Function $||\cdot ||:V\to [0,\infty )$ is called a norm if

1. $||x||=0$ if and only if $x=0$
2. $||\lambda x||=|\lambda |||x||$ for all $\lambda \in \mathbb{R}$, $x\in V$
3. $||x+y||\le ||x||+||y||$ for all $x,y\in V$

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Examples of Different Type of Norms

1. For $d_1$

$$||x|{|}_1:=\sum _{i=1}^n|x_i|$$

2. For $d_{\infty }$

$$||x|{|}_{\infty }:=\underset{i=1,\cdots ,n}{\max }|x_i|$$

3. For $l^p$ norms

$$||x|{|}_p:={\left(\sum _{i=1}^n|x_i{|}^p\right)}^{1/p}$$

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Relation between norm and metric spaces

Let $V$ be a vector space over the reals
Let $||\cdot ||$ be a norm on $V$
Define

$$d:V\times V\to [0,\infty )\text{ by }d(x,y):=||x-y||$$

Then

$$(V,d)\text{ is a metric space}$$

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Normed Space

Vector space endowed with norm $||\cdot ||$

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2.5 New Metric Spaces from Old Ones

Subspaces of Metric Spaces

Suppose that $(X,d)$ is a metric space
Let $Y$ be a subset of $X$ so that $Y\subset X$ then

The restriction of $d$ to $Y\times Y$ gives $Y$ a metric such that

$$(Y,d_{|Y\times Y})\text{ is a metric space}$$

Note if $X={\mathbb{R}}^n$ then $Y$ is just a subset not necessarily a subspace in terms of vector space

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Product Space

If $(X,d_X)$ and $(Y,d_Y)$ are metric spaces then $X\times Y$ can be a metric space

Suppose for $x_1,x_2\in X$ and $y_1,y_2\in Y$ then we have

$$d_{X\times Y}((x_1,y_1),(x_2,y_2))=\sqrt{d_X(x_1,x_2{)}^2+d_Y(y_1,y_2{)}^2}$$

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Product Space and Metrics lemma

Using same notation $d_{X\times Y}$ gives a metric on $X\times Y$

Proof

Positivity and Symmetry are obvious

Need to show that

$$\begin{array}{r}\sqrt{d_X(x_1,x_3{)}^2+d_Y(y_1,y_3{)}^2}+\sqrt{d_X(x_3,x_2{)}^2+d_Y(y_3,y_2{)}^2}\ge \\ \sqrt{d_X(x_1,x_2{)}^2+d_Y(y_1,y_2{)}^2}\end{array}$$

Let

$$\begin{array}{rl}{a}_1& =d_X(x_2,x_3)\\ a_2& =d_X(x_1,x_3)\\ a_3& =d_X(x_1,x_2)\\ & \\ b_1& =d_Y(y_2,y_3)\\ b_2& =d_Y(y_1,y_3)\\ b_3& =d_Y(y_1,y_2)\end{array}$$

So we need

$$\sqrt{a_2^2+b_2^2}+\sqrt{a_1^2+b_1^2}\ge \sqrt{a_3^2+b_3^2}$$

From triangle inequality we have

$$\begin{array}{rl}{a}_1+a_2& \ge a_3\\ b_1+b_2& \ge b_3\end{array}$$

Squaring and adding them we get

$$a_1^2+b_1^2+a_2^2+b_2^2+2(a_1{a}_2+b_1{b}_2)\ge a_3^2+b_3^2$$

By Cauchy-Schwarz then

$$a_1{a}_2+b_1{b}_2\le \sqrt{a_1^2+b_1^2}\sqrt{a_2^2+b_2^2}$$

Hence the result follows

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2.6 Balls and Boundedness

Open Ball

Let $X$ be a metric space

If $a\in X$ and $ϵ>0$ then the open ball of radius $ϵ$ about $\alpha$ is the set

$$B(a,ϵ)=\{x\in X:d(x,a)<ϵ\}$$

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Closed Ball

Let $X$ be a metric space

If $a\in X$ and $ϵ>0$ then the closed ball of radius $ϵ$ about $\alpha$ is the set

$$\underline{B}(a,ϵ)=\{x\in X:d(x,a)\le ϵ\}$$

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Bounded Metric Spaces

Let $X$ be a metric space and let $Y\subseteq X$

$Y$ is bounded if $Y$ is contained in some Open Ball

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Equivalent Properties for Bounded Metric Space lemma

Let $X$ be a metric space and let $Y\subseteq X$

1. $Y$ is bounded
2. $Y$ is contained in some closed ball
3. Set $\{d(y_1,y_2):y_1,y_2\in Y\}$ is a bounded subset of $\mathbb{R}$

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