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

\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}
\DeclareMathOperator*{\Ker}{Ker}
\DeclareMathOperator*{\Range}{Im}
\DeclareMathOperator*{\spn}{span}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{{\Large\bf Section 2.13:} \bf  1, A (optional)}
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}

\item Let $X$ and $Y$ be normed vector spaces over the same field, and
  let $A:X\to Y$ be a compact linear operator.  Show that $\Range A$
  is separable.
 
\item[A.] \textbf{(Optional)} Let $X$ be a normed vector space, $A:X\to X$ be a compact linear
  operator, and $\lambda\in\KK$ be non-zero.  Show that $\lambda I-A$
  is injective if and only if $\|(\lambda I-A)x_n\|\to \infty$ for
  every sequence $(x_n)_{n\in\NN}$ in $X$ satisfying $\|x_n\|\to\infty$.
  
\end{enumerate}


\end{document}