LEC ANA2 2

Example 162.1(Example).

If $(X,d)$ is a metric space, we can define a bounded metric

$$d_0(x,y)=\frac{d(x,y)}{1+d(x,y)}.$$

Convergence and open sets are preserved, so $d_0$ and $d$ are equivalent metrics. See @mathsstudent147If$dxy$Metric2020

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Examples of metrics on $C[0,1]$ ($\subset B[0,1]$), $C^1[0,1]$, $C^k[0,1]$, and $C^{\infty }[0,1]$.

$$\Vert f{\Vert }_{1,\infty }=\Vert f{\Vert }_{\infty }+\Vert f^{\prime }{\Vert }_{\infty }.$$ $$\Vert f{\Vert }_{k,\infty }=\Vert f{\Vert }_{\infty }+\sum _{i=1}^k\Vert f^{(k)}{\Vert }_{\infty }.$$ $$d(f,g)=\sum _{k=0}^{\infty }\frac{1}{k}\frac{\Vert f^{(k)}-g^{(k)}{\Vert }_{\infty }}{1+\Vert f^{(k)}-g^{(k)}{\Vert }_{\infty }}$$

^ this is not obtained from a norm.

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The $1$-norm on $\mathcal{R}[0,1]$ (Riemann integrable functions on $[0,1]$) is a pseudo norm: $\Vert f{\Vert }_1=0$ does not imply $f=0$.

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Defined a topology. Set of all open sets in a metric space is a topology.

Exercise: In $C[0,1]$, fix $f_0,g_0$. Show that $\{h:g_0(t)<h(t)<f_0(t)\forall t\}$ is open in $C[0,1]$.

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Proposition 162.2(Hausdorff metric).

Let $(X,d)$ be a metric space and let $\mathcal{K}$ denote the family of all non-empty bounded closed subsets of $X$. For $A,B\in \mathcal{K}$, let

$$\rho (A,B)=\inf \{ϵ>0:A\subseteq N_{ϵ}(B)\text{ and }B\subseteq N_{ϵ}(A)\},$$

where $N_{ϵ}(A)=\{x\in X:d(x,A)<ϵ\}$. $(\mathcal{K},\rho )$ is a metric space; $\rho$ is called the Hausdorff metric.

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