%% 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 17:36:42} %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 - Ot\U{e1}zky\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|} \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{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} \section{Ot\'{a}zky} Preverte si znalosti z\'{\i}skan\'{e} v tejto kapitole. Zodpovedajte v\v{s}% etky ot\'{a}zky. Ak neviete nejak\'{u} ot\'{a}zku zodpoveda\v{t} znovu pre% \v{s}tudujte pr\'{\i}slu\v{s}n\'{u} \v{c}as\v{t} a op\"{a}\v{t} odpovedajte. Po d\^{o}kladnom preveren\'{\i} Va\v{s}ich vedomost\'{\i} sa venujte po\v{c}% \'{\i}taniu pr\'{\i}kladov. \begin{enumerate} \item Definujte skal\'{a}rny s\'{u}\v{c}in\label{8} prvkov $\mathbf{x},% \mathbf{y\in R}^{n}.$ \item Definujte normu prvku $\mathbf{x\in R}^{n}.$ \item Formulujte Schwarzovu nerovnos\v{t}. \item Nap\'{\i}\v{s}te vlastnosti normy v $\mathbf{R}^{n}.$ \item Definujte metriku v $\mathbf{R}^{n}.$ \item Definujte otvoren\'{e} okolie bodu v $\mathbf{R}^{n}.$ \item Kedy hovor\'{\i}me, \v{z}e bod $\mathbf{x}$ je vn\'{u}torn\'{y}m bodom mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Kedy hovor\'{\i}me, \v{z}e bod $\mathbf{x}$ je hrani\v{c}n\'{y}m bodom mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Kedy hovor\'{\i}me, \v{z}e bod $\mathbf{x}$ je vonkaj\v{s}\'{\i}m bodom mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Definujte vn\'{u}tro mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Definujte hranicu mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Definujte vonkaj\v{s}ok mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Definujte uz\'{a}ver mno\v{z}iny $A\subset \mathbf{R}^{n}.$ \item Definujte uzavret\'{u} mno\v{z}inu. \item Plat\'{\i} nasleduj\'{u}ce tvrdenie: ak $A\subset \mathbf{R}^{n},$ potom je mno\v{z}ina $A$ bu\v{d} uzavret\'{a}, alebo otvoren\'{a}? \item Kedy je bod $\mathbf{a}\in \mathbf{R}^{n}$ hromadn\'{y} bod mno\v{z}% iny $A\subset \mathbf{R}^{n}?$ \item Kedy je bod $\mathbf{a}\in \mathbf{R}^{n}$ izolovan\'{y} bod mno\v{z}% iny $A\subset \mathbf{R}^{n}?$ \item Nech $A=\mathbf{R}^{n}\setminus \left\{ \mathbf{0}\right\} .$ Bod $% \mathbf{a}=\mathbf{0}$ je hromadn\'{y} alebo izolovan\'{y} bod mno\v{z}iny $% A?$ \item Nech $\lim_{k\longrightarrow \infty }\mathbf{x}^{(k)}=\mathbf{x,}$ ak% \'{e} tvrdenie plat\'{\i} pre komponenty $\mathbf{x}^{(k)}$ a $\mathbf{x?}$ \item Plat\'{\i} tvrdenie: ak je postupnos\v{t} $\{\mathbf{x}% ^{(k)}\}_{k=1}^{\infty }$ ohrani\v{c}en\'{a}, potom je aj konvergentn\'{a}? \item Plat\'{\i} tvrdenie: nech $\lim_{k\longrightarrow \infty }\mathbf{x}% ^{(k)}=\mathbf{x}$ \ pri\v{c}om $\mathbf{x}^{(k)}\in A\ \forall \ k\in \mathbf{N,}$ potom $\mathbf{x}\in \overline{A}?$ \item Definujte kompaktn\'{u} mno\v{z}inu. \end{enumerate} \begin{center} \begin{tabular}{|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{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}