\documentclass[12pt]{article}

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\usepackage{fullpage,amsmath,amssymb,amsthm,framed,graphicx,color}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\usepackage{pifont}% http://ctan.org/pkg/pifont
\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%

\def\NN{\mathbb{N}}
\def\ZZ{\mathbb{Z}}
\def\RR{\mathbb{R}}
\def\QQ{\mathbb{Q}}
\def\CC{\mathbb{C}}
\def\KK{\mathbb{K}}

\def\cO{\mathcal{O}}
\def\cP{\mathcal{P}}
\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cF{\mathcal{F}}
\def\cH{\mathcal{H}}
\def\cR{\mathcal{R}}
\def\cN{\mathcal{N}}
\def\cL{\mathcal{L}}

\def\ee{\varepsilon}
\def\ii{\mathfrak{i}}

\newcommand{\mb}[1]{\ensuremath{\mathbf{ #1 }}}
\newcommand{\norm}[1]{\ensuremath{\left\| #1\right\| }}
\newcommand{\enorm}[1]{\ensuremath{|\!|\!| #1|\!|\!| }}

\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator*{\supp}{supp}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{{\Large\bf Section 2.10:} \bf 1, 2 (optional), A (optional) }
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}


  
\item[1.] Let $Y$ be a finite dimensional subspace of a normed vector
  space $(X,\|\cdot\|)$.
  \begin{enumerate}
  \item Show that, for any $x\in X$, there is a $\tilde{y}\in Y$ (not
    necessarily unique) such that $\displaystyle
    \|x-\tilde{y}\|=\inf_{y\in Y}\|x-y\|$.

  
  \item Suppose that $(X,\|\cdot\|)$ is \textit{strictly convex},
    i.e. whenever $u,v\in X$ are distinct and satisfy $\|u\|=\|v\|=1$,
    it follows that $\|u/2+v/2\|<(1/2)+(1/2)=1$.  Show that the vector
    $\tilde{y}\in Y$ found above is necessarily unique in this case.
 
\item Let $n\in\NN$, $n\geq 2$.  Show that $(\KK^n,\|\cdot\|_p)$ is strictly convex for any
  $p\in(1,\infty)$, but not for $p=1$ or $p=\infty$.
  
  \end{enumerate}

\item[2.] \textbf{(Optional)}  Show that the interior of a compact subset of an infinite
dimensional normed vector space must be empty.

\item[A.] \textbf{(Optional)} Let $(X,\|\cdot\|)$ be a finite dimensional normed vector
  space.  Show that any bounded sequence in $(X,\|\cdot\|)$ must have
  a convergent subsequence.
    
\end{enumerate}


\end{document}


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