\relax 
\providecommand\zref@newlabel[2]{}
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {section}{\numberline {1}Lecture 8 -- Lp Spaces and Classical Inequalities}{1}{section.1}\protected@file@percent }
\newlabel{sec:lec08}{{1}{1}{Lecture 8 -- Lp Spaces and Classical Inequalities}{section.1}{}}
\newlabel{sec:lec08@cref}{{[section][1][]1}{[1][1][]1}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}The spaces $L^p$}{1}{subsection.1.1}\protected@file@percent }
\newlabel{def:Lp-finite}{{1.1}{1}{$L^p$ space, $1\le p<\infty $}{tcb@cnt@definition.1.1}{}}
\newlabel{def:Lp-finite@cref}{{[tcb@cnt@definition][1][1]1.1}{[1][1][]1}{}{}{}}
\newlabel{def:Linfty}{{1.2}{1}{Essential supremum and $L^\infty $}{tcb@cnt@definition.1.2}{}}
\newlabel{def:Linfty@cref}{{[tcb@cnt@definition][2][1]1.2}{[1][1][]1}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Markov, Chebyshev, Chernoff}{1}{subsection.1.2}\protected@file@percent }
\newlabel{thm:markov}{{1.3}{1}{Markov's inequality}{tcb@cnt@definition.1.3}{}}
\newlabel{thm:markov@cref}{{[tcb@cnt@definition][3][1]1.3}{[1][1][]1}{}{}{}}
\newlabel{cor:chebyshev}{{1.4}{2}{Chebyshev's inequality}{tcb@cnt@definition.1.4}{}}
\newlabel{cor:chebyshev@cref}{{[tcb@cnt@definition][4][1]1.4}{[1][1][]2}{}{}{}}
\newlabel{cor:chernoff}{{1.5}{2}{Chernoff's inequality}{tcb@cnt@definition.1.5}{}}
\newlabel{cor:chernoff@cref}{{[tcb@cnt@definition][5][1]1.5}{[1][2][]2}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Convexity and Jensen's inequality}{2}{subsection.1.3}\protected@file@percent }
\newlabel{def:convex}{{1.6}{2}{Convex function}{tcb@cnt@definition.1.6}{}}
\newlabel{def:convex@cref}{{[tcb@cnt@definition][6][1]1.6}{[1][2][]2}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A convex function: the secant joining \((x,\phi (x))\) and \((y,\phi (y))\) lies above the graph.}}{2}{figure.caption.1}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:convex}{{1}{2}{A convex function: the secant joining \((x,\phi (x))\) and \((y,\phi (y))\) lies above the graph}{figure.caption.1}{}}
\newlabel{fig:convex@cref}{{[figure][1][]1}{[1][2][]2}{}{}{}}
\newlabel{thm:jensen}{{1.7}{2}{Jensen's inequality}{tcb@cnt@definition.1.7}{}}
\newlabel{thm:jensen@cref}{{[tcb@cnt@definition][7][1]1.7}{[1][2][]2}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}H\"older and Minkowski}{3}{subsection.1.4}\protected@file@percent }
\newlabel{thm:holder}{{1.8}{3}{H\"older's inequality}{tcb@cnt@definition.1.8}{}}
\newlabel{thm:holder@cref}{{[tcb@cnt@definition][8][1]1.8}{[1][3][]3}{}{}{}}
\newlabel{cor:cauchy-schwarz}{{1.9}{3}{Cauchy--Schwarz}{tcb@cnt@definition.1.9}{}}
\newlabel{cor:cauchy-schwarz@cref}{{[tcb@cnt@definition][9][1]1.9}{[1][3][]3}{}{}{}}
\newlabel{thm:minkowski}{{1.10}{3}{Minkowski's inequality}{tcb@cnt@definition.1.10}{}}
\newlabel{thm:minkowski@cref}{{[tcb@cnt@definition][10][1]1.10}{[1][3][]3}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5}Approximation in $L^p$}{3}{subsection.1.5}\protected@file@percent }
\newlabel{thm:Lp-density}{{1.11}{4}{Density of simple functions in $L^p$}{tcb@cnt@definition.1.11}{}}
\newlabel{thm:Lp-density@cref}{{[tcb@cnt@definition][11][1]1.11}{[1][3][]4}{}{}{}}
\gdef \@abspage@last{4}