07 - Metric Space

Metric Space

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

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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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

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

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

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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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

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 $\mathbb{R}$