\input /home/pasky/school/fastex/lib.tex


\subchapter{�vod}{

\scpart{Motivace}{

	M�me nepr�zdnou mno�inu $P$
	{\it (mno�inu �ehokoliv --- m��e j�t o mno�inu v�ech �iraf v Ma�arsku,
	mno�inu v�ech pravdiv�ch v�rok� Stanislava Grosse --- nebo i n�co
	v�t��ho)},
	mezi jej�mi� objekty chceme m��it vzd�lenost.

}

\definition{
	Bu� $P$ nepr�zdn� mno�ina a $\ro\colon P \times P \to [0,\infty)$.
	Jestli�e plat�:
	\list{
	\listItem{$\ro(x,y)=0 \Leftrightarrow x=y$ \hfill $\qquad\forall x,y \in P$}
	\listItem{$\ro(x,y)=\ro(y,x)$ \hfill {\sl (symetrie)} $\qquad\forall x,y \in P$}
	\listItem{$\ro(x,y) \le \ro(x,z)+\ro(z,y)$ \hfill {\sl (troj�heln�kov� nerovnost)} $\qquad\forall x,y \in P$}
	}
	Potom $\ro$ je {\bf metrika} na $P$
	a $(P,\ro)$ naz�v�me {\bf metrick�m prostorem}.
}

\shortexerc{Je v metrick�m prostoru n�dra�� cena l�stku metrikou? \DCV}

\examples{
\list{
\listItem{$P = \real$, $\ro = \ro_{\rm eukl.}(x,y) := |x-y|$}
\listItem{$P \subset \real$, $\ro = \ro_{\rm eukl.}(x,y)$ (podprostor se zd�d�nou metrikou)}
\listItem{$P = \real^n = \{x=(x_1,x_2,\ldots,x_n),\ x_j \in \real,\ j = 1,\ldots,n\}$, $n \in \nat$

	\list{
	\alistItem{$P = \real^n$, \M{\ro = \ro_{\rm eukl.}(x,y) := \sqrt{\sum_{j=1}^n (y_j-x_j)^2}}

		(i) a (ii) trivi�ln�. Doka�me si (iii):
		$$ \sqrt{\sum_{j=1}^n (y_j-x_j)^2} \le \sqrt{\sum_{j=1}^n (x_j-z_j)^2} + \sqrt{\sum_{j=1}^n (z_j-y_j)^2} $$

		Ozna�me $a_j := x_j - z_j$, $b_j := z_j - y_j$, tedy chci
		$$ \sqrt{\sum (a_j+b_j)^2} \le \sqrt{\sum a_j^2} + \sqrt{\sum b_j^2} $$
		$$ \eqalign{\sqrt{\sum (a_j+b_j)^2}
			    & \becauseof{\phantom{\ref{Cauchy}}}{=} \sqrt{\sum a_j^2 + 2\sum a_jb_j + \sum b_j^2} \cr
			    & \becauseof{\ref{Cauchy}}{\le}
				 \sqrt{\sum a_j^2 + 2\sqrt{\sum a_j^2}\sqrt{\sum b_j^2} + \sum b_j^2}
				 = \sqrt{\sum a_j^2} + \sqrt{\sum b_j^2}} $$
	}
	\alistItem{newyorksk� metrika: \M{\ro_{\rm NY}(x,y) := \sum_{j=1}^n |x_j-y_j|}}
	\alistItem{maximov� metrika: \M{\ro_{\rm max}(x,y) := \max_{j=1,\ldots,n} |x_j-y_j|}}
	\alistItem{``rusk�'' metrika: m�jme $a \in P$,
		\M{\ro_{\rm pamp.}(x,y) := \cases{0 & $x=y$\cr \ro(x,a) + \ro(a,y) & $x \ne y$}}}
	}
}
\penalty-100
\listItem{$P = C[a,b] = \{\hbox{$f$ definovan� a spojit� na $[a,b]$}\}$

	\list{
	\alistItem{integr�ln� metrika: \M{\ro_{\rm int}(f,g) := \int_a^b |f(x)-g(x)|\,dx}}
	\alistItem{suprem�ln� metrika: \M{\ro_{\rm sup}(f,g) := \sup_{x\in[a,b]} |f(x)-g(x)|}
	
		\smallskip
		Z obr�zku vypl�v�, �e $\ro_{\rm int}(f,g) \le (b-a)\ro_{\rm sup}(f,g)$}
	}
}
\listItem{diskr�tn� metrika: $P$ je cokoliv, $\ro_{\rm diskr.}(x,y) = \cases{0 & $x=y$\cr1 & $x\ne y$}$}
}
}


\definition{
	$(P,\ro)$ bu� metrick� prostor, $x \in P$, $r > 0$.
	Pak {\bf otev�enou koul�} se st�edem $x$ a polom�rem $r$
	rozum�me mno�inu $$B(x,r) := \{y \in P: \ro(x,y)<r\}$$
	Obdobn� definujeme {\bf uzav�enou kouli} jako
	$$\overline{B}(x,r) := \{y \in P: \ro(x,y) \le r\}$$

	Pozor, nap�. kdy� $P = [0,1]$, $x = 1/4$, $r = 1/2$, pak
	$B(x,r) = B(1/4,1/2) = [0,3/4)$!
}


\examples{
	M�jme $P = \real^2$, $x = [0,0]$, $r=1$ (jednotkov� koule se st�edem 0: $B([0,0],1)$).

	Spousta obr�zk�...
	\list{
	\nlistItem{$\ro_{\rm eukl.}$: kole�ko}
	\nlistItem{$\ro_{\rm NY}$: koso�tverec}
	\nlistItem{$\ro_{\rm max}$: �tvere�ek}
	\nlistItem{$\ro_{\rm diskr.}$: \M{B(x,r) = \cases{\{x\} & $r \le 1$ \cr P & $r > 1$}}}
	\nlistItem{$\ro_{\rm pamp.}$: \DCV}
	}

	M�jme $P = C[a,b]$, $f$, $r>0$.

	\list{
	\nlistItem{$\ro_{\rm sup}$: $r$-ov� p�sek kolem funkce, v�echny funkce, kter� z n�j nevylezou}
	\nlistItem{$\ro_{\rm int}$: nelze nakreslit}
	}
}


\definition{
	$(P,\ro)$ bu� metrick� prostor a $G \subseteq P$.
	�ekneme, �e $G$ je {\bf otev�en� v $P$}, pokud
	$$ \forall x \in G\ \existss r > 0: B(x,r) \subseteq G $$

	Naopak $G$ je {\bf uzav�en� v $P$},
	pokud $P \setminus G$ je otev�en�.
}



\proof{
\list{
\listItem{z�ejm�}
\listItem{$F \supset B$, $F$ uz., tedy \M{F \supset A \Rightarrow \bigcap_{F \supset B} F \supset \bigcap_{F \supset A} F}}
\listItem{$\overline{A}$ je uz. a $A \subset \overline{A}$ $\Rightarrow \overline{\overline{A}} = \overline{A}$}
\listItem{
\list{
\namedListItem{``$\subseteq$''}{}
}
}
}
}

}


\bend
