%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:45:49} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Metrick\U{e9} priestory - Line\U{e1}rne a metrick\U{e9} priestory\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Metrick\'{e} priestory} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma1.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma12.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma12.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O1.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C1.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \paragraph{Ciele} Po pre\v{s}tudovan\'{\i} tejto \v{c}asti by ste mali by\v{t} schopn\'{\i}: \begin{itemize} \item ovl\'{a}da\v{t} z\'{a}kladn\'{e} vlastnosti priestorov $\mathbf{R}% ^{n}, $\ alebo $\mathbf{V}\left( \mathbf{R}^{n}\right) $ ako aj line\'{a}% rnych a metrick\'{y}ch priestorov, \item op\'{\i}sa\v{t} d\^{o}vody pre\v{c}o boli zaveden\'{e} pojmy vn\'{u}% torn\'{e}ho, vonkaj\v{s}ieho, hrani\v{c}n\'{e}ho, hromadn\'{e}ho a izolovan% \'{e}ho bodu mno\v{z}iny, \item vysvetli\v{t} pojmy otvoren\'{a} mno\v{z}ina, uzavret\'{a} mno\v{z}ina, \item aplikova\v{t} znalosti o postupnostiach v priestoroch $\mathbf{R}^{n}$% \ pri chrakterizovan\'{\i} r\^{o}znych typov konvergenci\'{\i}. \end{itemize} \paragraph{Po\v{z}adovan\'{e} vedomosti:} \begin{itemize} \item znalos\v{t} z\'{a}kladn\'{y}ch typov priestorov z line\'{a}rnej algebry, \item znalos\v{t} element\'{a}rnych pojmov z matematickej anal\'{y}zy I.. \end{itemize} \subsection{\protect\bigskip Line\'{a}rne a metrick\'{e} priestory.} \subsubsection{Line\'{a}rne priestory.} V tejto kapitole sa budeme zaobera\v{t} z\'{a}kladn\'{y}mi vlastnos\v{t}ami line\'{a}rnych priestorov, ktor\'{e} je nutn\'{e} ovl\'{a}da\v{t} pri \v{s}t% \'{u}diu diferenci\'{a}lneho a integr\'{a}lneho po\v{c}tu funkci\'{\i} viacer% \'{y}ch premenn\'{y}ch. V matematickej anal\'{y}ze I sme sa zaoberali s vlastnos\v{t}ami mno\v{z}iny re\'{a}lnych \v{c}\'{\i}sel $\mathbf{R.}$Re\'{a}% lne \v{c}\'{\i}sla budeme teraz naz\'{y}va\v{t} \emph{skal\'{a}ry}\textit{. }% Nech $n>1,\,n\in \mathbf{N,}$ ozna\v{c}me \emph{mno\v{z}inu v\v{s}etk\'{y}ch usporiadan\'{y}ch n-t\'{\i}c re\'{a}lnych \v{c}\'{\i}sel:} \[ \mathbf{R}^{n}=\left\{ \left( x_{1},x_{2},...,x_{n}\right) :x_{i}\in \mathbf{% R,\,\forall }i=1,2,...,n\right\} . \]% Prvky mno\v{z}iny $\mathbf{R}^{n}$\ budeme ozna\v{c}ova\v{t} tu\v{c}n\'{y}mi p\'{\i}smenami $\mathbf{x,\,y,\,z,....}$ Ak $\mathbf{x}\in \mathbf{R}^{n},\,% \mathbf{x}=\left( x_{1},x_{2},...,x_{n}\right) ,\,$re\'{a}lne \v{c}\'{\i}slo $x_{i}$ \ naz\'{y}vame \emph{i-ta komponenta} prvku $\mathbf{x.}$ Ozna\v{c}% me prvok, ktor\'{e}ho komponenty s\'{u} nuly $\mathbf{0}=\left( 0,0,...,0\right) \in \mathbf{R}^{n}.$ Pre $n=2$ namiesto $\mathbf{x}=\left( x_{1},x_{2}\right) $ oby\v{c}ajne p\'{\i}\v{s}eme $\mathbf{x}=\left( x,y\right) ,$ podobne pre $n=3$ miesto $\mathbf{x}=\left( x_{1},x_{2},x_{3}\right) $ p\'{\i}\v{s}eme $\mathbf{x}=\left( x,y,z\right) .$ Pre $n=2$ prvok $\mathbf{x}=(x_{1},x_{2})$ m\^{o}\v{z}me pova\v{z}ova\v{t} za bod v rovine s kart\'{e}zskymi s\'{u}radnicami $(x_{1},x_{2})$. Podobne prvok $\mathbf{x}\in \mathbf{R}{^{3}}$ m\^{o}\v{z}me stoto\v{z}ni\v{t} s bodom trojrozmern\'{e}ho priestoru. \v{T}a\v{z}\v{s}ie sa daj\'{u} predstavi% \v{t} body z priestoru ${\ \mathbf{R}^{n}}$, pre $n>3$. Pre prvky z ${\ \mathbf{R}^{n}}$ definujeme \[ \mathbf{x}=\mathbf{y}\ \Leftrightarrow \ x_{i}=y_{i},\ \forall i=1,2,\dots ,n. \]% Ke\v{d} v mno\v{z}ine ${\mathbf{R}^{n}}$ definujeme oper\'{a}cie \emph{s\v{c}% \'{\i}tania} dvoch prvkov a \emph{n\'{a}sobenia prvku skal\'{a}rom} po zlo% \v{z}k\'{a}ch, t.j. ak $\mathbf{x,\,y}\in \mathbf{R}{^{n},\,\alpha \in }% \mathbf{R,}$ tak \[ \mathbf{x}+\mathbf{y}=(x_{1}+y_{1},x_{2}+y_{2},\dots ,x_{n}+y_{n})\in \mathbf{R}{^{n}} \]% \[ \alpha \mathbf{x}=(\alpha x_{1},\alpha x_{2},\dots ,\alpha x_{n})\in \mathbf{% R}{^{n}}, \]% pre rozdiel \[ \mathbf{x}-\mathbf{y}=\mathbf{x}+(-1)\mathbf{y}, \]% potom $\mathbf{R}^{n}$ s oper\'{a}ciou s\v{c}\'{\i}tania dvoch prvkov a n% \'{a}sobenia prvku skal\'{a}rom je \emph{line\'{a}rny (vektorov\'{y}) priestor\label{3}} nad $\mathbf{R.}$ Line\'{a}rne priestory sa \v{s}tuduj\'{u} v line\'{a}rnej algebre, odkia\v{l} tie\v{z} vieme, \v{z}e line\'{a}rny priestor $\mathbf{R}^{n}$ m\'{a} \emph{% dimenziu\label{5}} $n,$ to znamen\'{a}, \v{z}e existuje kone\v{c}n\'{a} $n$% \emph{-prvkov\'{a} line\'{a}rne nez\'{a}visl\'{a} mno\v{z}ina prvkov} $\{% \mathbf{e}_{1},\mathbf{e}_{2},\dots ,\mathbf{e}_{n};\,\mathbf{e}_{i}\in \mathbf{R}^{n}\}$ minim\'{a}lna v tom zmysle, \v{z}e nem\^{o}\v{z}e ma\v{t} menej ako $n$ prvkov, ktor\'{u} naz\'{y}vame \emph{b\'{a}za\label{6}} line% \'{a}rneho priestoru $\mathbf{R}^{n}$, tak\'{a} \v{z}e $\forall \ \mathbf{x}% \in \mathbf{R}^{n},\ \mathbf{x}=x_{1}\mathbf{e}_{1}+x_{2}\mathbf{e}% _{2}+\dots +x_{n}\mathbf{e}_{n}$ a komponenty $x_{i}\in \mathbf{R},\ \forall i=1,\dots ,n$ s\'{u} definovan\'{e} jednozna\v{c}ne. \v{D}alej budeme pou% \v{z}\'{\i}va\v{t} \v{s}tandardn\'{u} b\'{a}zu: $\mathbf{e}_{1}=\left( 1,0,\dots ,0\right) ,\,\mathbf{e}_{2}=\left( 0,1,\dots ,0\right) ,\dots ,% \mathbf{e}_{n}=\left( 0,0,\dots ,1\right) .$ Ak $A,B$ s\'{u} dve mno\v{z}iny, potom \emph{kart\'{e}zskym s\'{u}\v{c}inom mno\v{z}\'{\i}n\label{7} }$A,\,B$ rozumieme mno\v{z}inu \[ A\times B=\{(a,b);a\in A,b\in B\}, \]% podobne \[ A_{1}\times A_{2}\times \dots \times A_{m}=\{(a_{1},a_{2},\dots ,a_{m});a_{i}\in A_{i},\ \forall i=1,2,\dots ,m\}. \]% Potom napr\'{\i}klad \[ \mathbf{R}^{2}\simeq \mathbf{R}\times \mathbf{R} \]% a \[ \mathbf{R}^{n}\times \mathbf{R}^{m}\simeq \mathbf{R}^{n+m}. \] Preto\v{z}e matematick\'{a} anal\'{y}za sa zaober\'{a} najm\"{a} pojmami \emph{spojitosti} a \emph{diferencovate\v{l}nosti funkci\'{\i}, }d\^{o}le% \v{z}it\'{y} je pojem \textsl{limity.} V $\mathbf{R}^{n}$ pojem limity definujeme pomocou \emph{vzdialenosti (metriky) }dvoch prvkov z $\mathbf{R}% ^{n}$ a vzdialenos\v{t} definujeme pomocou \emph{skal\'{a}rneho s\'{u}\v{c}% inu}. \begin{definition} \emph{Skal\'{a}rny s\'{u}\v{c}in}\label{8} prvkov $\mathbf{x},\mathbf{y\in R}% ^{n}$ je zobrazenie \[ .\cdot .:\mathbf{R}^{n}\times \mathbf{R}^{n}\rightarrow \mathbf{R} \]% \[ \mathbf{x}\cdot \mathbf{y}=\sum_{i=1}^{n}x_{i}y_{i} \]% s vlastnos\v{t}ami% \begin{equation} \mathbf{x\cdot x}\geq 0,\ \text{kde rovnos\v{t} nastane vtedy a len vtedy ak}% \ \mathbf{x}=\mathbf{0,} \tag{(i)} \end{equation}% \begin{equation} \left( \mathbf{x}+\mathbf{y}\right) \cdot \mathbf{z}=\mathbf{x\cdot z}+% \mathbf{y\cdot z}, \tag{(ii)} \end{equation}% \begin{equation} \left( \alpha \mathbf{x}\right) \mathbf{\cdot y}=\alpha \mathbf{x\cdot y},\ \forall \alpha \in \mathbf{R,} \tag{(iii)} \end{equation}% \begin{equation} \mathbf{x\cdot y}=\mathbf{y\cdot x}. \tag{(iv)} \end{equation} \end{definition} \begin{definition} \emph{Normou}\label{9} prvku $\mathbf{x}\in \mathbf{R}^{n}$ naz\'{y}vame nez% \'{a}porn\'{e} re\'{a}lne \v{c}\'{\i}slo% \[ \Vert \mathbf{x}\Vert =\sqrt{\mathbf{x\cdot x}}=\sqrt{\sum_{i=1}^{n}x_{i}^{2}% }. \] \end{definition} Ak $n=1$, normou prvku $x=\mathbf{x}\in \mathbf{R}$ je $\Vert \mathbf{x}% \Vert =\sqrt{\mathbf{x\cdot x}}=\sqrt{x_{{}}^{2}}=|x|,$ t.j. vzdialenos\v{t} bodu od za\v{c}iatku s\'{u}radnicovej s\'{u}stavy. Podobne v priestoroch $% \mathbf{R}^{2}$ a $\mathbf{R}^{3}$ norma prvku znamen\'{a} jeho vzdialenos% \v{t} od za\v{c}iatku s\'{u}radnicovej s\'{u}stavy (pod\v{l}a Pytagorovej vety). Tak\'{u} ist\'{u} interpret\'{a}ciu normy budeme ma\v{t} na mysli aj pre priestory $\mathbf{R}^{n}$ a dan\'{u} normu budeme vola\v{t} \emph{% euklidovskou normou} prvku $\mathbf{x}$. \CustomNote{Note}{% Skal\'{a}rny s\'{u}\v{c}in v priestoroch $\mathbf{R}^{3}(\mathbf{R}^{2})$ m% \^{o}\v{z}eme interpretova\v{t} ako mieru uhla $\theta $ medzi prvkami $% \mathbf{x,\,y}$. Naozaj v $\mathbf{R}^{3}(\mathbf{R}^{2})$ plat\'{\i} \[ \mathbf{x\cdot y}=\Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert \cos {\theta }% ,\ (0\leq \theta \leq \pi ), \]% aj kos\'{\i}nusov\'{a} veta \[ \Vert \mathbf{x}-\mathbf{y}\Vert ^{2}=\Vert \mathbf{x}\Vert ^{2}+\Vert \mathbf{y}\Vert ^{2}-2\Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert \cos {% \theta }. \]% } \ Nech pre $\mathbf{x},\mathbf{y}\in \mathbf{R}^{n}$ plat\'{\i} $\mathbf{% x\cdot y}=0,$ vtedy hovor\'{\i}me, \v{z}e prvky $\mathbf{x}$ a $\mathbf{y}$ s% \'{u} \emph{ortogon\'{a}lne.\label{10}} Pre \v{l}ubovo\v{l}n\'{e} ortogon% \'{a}lne vektory $\mathbf{x},\mathbf{y}\in \mathbf{R}^{n}$ \v{l}ahko mo\v{z}% no overi\v{t} platnos\v{t} rovnosti: \[ \Vert \mathbf{x}+\mathbf{y}\Vert ^{2}=\Vert \mathbf{x}\Vert ^{2}+\Vert \mathbf{y}\Vert ^{2}\ \]% zn\'{a}mej pod n\'{a}zvom \emph{Pytagorova veta}, pre ka\v{z}d\'{e} $\mathbf{% x},\mathbf{y\in R}^{n}$ plat\'{\i} \[ \Vert \mathbf{x}+\mathbf{y}\Vert ^{2}+\Vert \mathbf{x}-\mathbf{y}\Vert ^{2}=2\Vert \mathbf{x}\Vert ^{2}+2\Vert \mathbf{y}\Vert ^{2}\ \]% zn\'{a}ma pod n\'{a}zvom \emph{rovnobe\v{z}n\'{\i}kov\'{a} rovnos\v{t}}. \begin{theorem} (Schwarzova nerovnos\v{t} \label{11}) Pre ka\v{z}d\'{e} $\mathbf{x},\mathbf{y% }\in \mathbf{R}^{n}$ plat\'{\i} \begin{equation} |\mathbf{x\cdot y}|\leq \Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert . \tag{(Schwarzova nerovnos\U{165})} \end{equation} \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO111.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO111.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{theorem} Pre ka\v{z}d\'{e} $\mathbf{x},\mathbf{y}\in \mathbf{R}^{n}$ a $\alpha \in \mathbf{R}$ plat\'{\i} \begin{equation} \Vert \mathbf{x}\Vert \geq 0\ \text{a rovnos\v{t} nast\'{a}va}% \Leftrightarrow \ \text{ak}\ \mathbf{x}=\mathbf{0} \tag{(a)} \end{equation}% \begin{equation} \Vert \alpha \mathbf{x}\Vert =|\alpha |\Vert \mathbf{x}\Vert \tag{(b)} \end{equation}% \begin{equation} \Vert \mathbf{x-y}\Vert =\Vert \mathbf{y-x}\Vert \tag{(c)} \end{equation}% \begin{equation} \Vert \mathbf{x+y}\Vert \leq \Vert \mathbf{x}\Vert +\Vert \mathbf{y}\Vert \ -\ \text{trojuholn\'{\i}kov\'{a} nerovnos\v{t}\label{12}} \tag{(d)} \end{equation}% \begin{equation} |\Vert \mathbf{x}\Vert -\Vert \mathbf{y}\Vert |\leq \Vert \mathbf{x-y}\Vert . \tag{(e)} \end{equation} \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO112.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO112.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} \subsubsection{\protect\bigskip Metrick\'{e} priestory.} \begin{definition} Nech $X\neq \emptyset $ je mno\v{z}ina. Definujeme zobrazenie $\rho :X\times X\longrightarrow \mathbf{R,}$ ktor\'{e} m\'{a} vlastnosti: \begin{equation} \left. \begin{tabular}{c} $\rho \left( x,y\right) \geq 0$ a $\rho \left( x,y\right) =0\,\Longleftrightarrow x=y,$ \\ $\rho \left( x,y\right) =\rho \left( y,x\right) ,$ \\ $\rho \left( x,z\right) \leq \rho \left( x,y\right) +\rho \left( y,z\right) $% \end{tabular}% \right\} \tag{(m)} \end{equation}% pre ka\v{z}d\'{e} $x,y,z\in X.$ Potom $\left( X,\rho \right) $ naz\'{y}vame \emph{metrick\'{y} priestor\label{13}} s metrikou $\rho .$ \v{C}\'{\i}slo $% \rho \left( x,y\right) $ sa naz\'{y}va \emph{vzdialenos\v{t}ou\label{14} }% prvkov $x$ a $y.$ \end{definition} Ak $X=\mathbf{R}^{n}$ a polo\v{z}\'{\i}me $\rho \left( \mathbf{x},\mathbf{y}% \right) =\left\| \mathbf{x}-\mathbf{y}\right\| ,$ s euklidovskou normou, potom vlastnosti metriky $\rho $ \ - (m) s\'{u} splnen\'{e} a priestor $% \left( \mathbf{R}^{n},\rho \right) $ s metrikou $\rho $ je metrick\'{y} priestor. $\mathbf{R}^{n}$ m\^{o}\v{z}me uva\v{z}ova\v{t} ako \begin{itemize} \item \quotedblbase bodov\'{y} euklidovsk\'{y} priestor'', ke\v{d} uva\v{z}% ujeme iba euklidovsk\'{u} metriku, t.j. \v{l}ubovo\v{l}n\'{e} dva prvky $% \mathbf{x},\mathbf{y}\in \mathbf{R}^{n}$ pova\v{z}ujeme za body (ako v analytickej geometrii) a ich vzdialenos\v{t}ou rozumieme euklidovsk\'{u} normu ich rozdielu (v tomto pr\'{\i}pade budeme \ pre $\left( \mathbf{R}% ^{n},\rho \right) $\ pou\v{z}\'{\i}va\v{t} ozna\v{c}enie $\mathbf{R}^{n}$\ a \v{s}tandardn\'{e} ozna\v{c}enie prvku $\mathbf{x}\in \mathbf{R}^{n},\,% \mathbf{x}=\left( x_{1},x_{2},...,x_{n}\right) $ ), \item \quotedblbase vektorov\'{y} euklidovsk\'{y} priestor'', ke\v{d} prvky z $\mathbf{R}^{n}$ pova\v{z}ujeme za vektory a $\mathbf{R}^{n}$ uva\v{z}% ujeme nielen s euklidovskou normou, ale aj s vektorov\'{y}mi oper\'{a}ciami (v tomto pr\'{\i}pade budeme \ pre $\left( \mathbf{R}^{n},\rho \right) $\ pou% \v{z}\'{\i}va\v{t} ozna\v{c}enie $\mathbf{V}\left( \mathbf{R}^{n}\right) $\ a \v{s}tandardn\'{e} ozna\v{c}enie prvku $\mathbf{x}\in \mathbf{V}\left( \mathbf{R}^{n}\right) ,\,\mathbf{x}=x_{1}\mathbf{e}_{1}+x_{2}\mathbf{e}% _{2}+...+x_{n}\mathbf{e}_{n}$). V pr\'{\i}pade vektorov\'{e}ho priestoru $% \mathbf{V}\left( \mathbf{R}^{3}\right) $\ ozna\v{c}ujeme \v{s}tandardn\'{u} b% \'{a}zu: $\mathbf{e}_{1}=\mathbf{i},\,\mathbf{e}_{2}=\mathbf{j},\,\mathbf{e}% _{3}=\mathbf{k.}\,$ \end{itemize} Budeme pou\v{z}\'{\i}va\v{t} interpret\'{a}ciu, ktor\'{a} bude v danom pr% \'{\i}pade vhodnej\v{s}ia. Znalosti z line\'{a}rnej algebry n\'{a}m toto nazeranie umo\v{z}nia. \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma1.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma12.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma12.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O1.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C1.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C1.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Metrick\'{e} priestory} \end{document}