\documentclass[12pt]{article}

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\usepackage{fullpage,amsmath,amssymb,graphicx,color}
\usepackage{amsthm}

\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}
\newtheorem{theorem}{Theorem}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{\bf{\Large\bf Section 3.6:} 
  A, 3}
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}

\item[A.] Let $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ be normed
  spaces, and suppose that $A\in\cL(X,Y)$, $B,C\in\cL(Y,X)$ and $M,N\in\cL(X)$.  Show
  the following:
  \begin{enumerate}
  \item If $AB=I_Y$ (the identity map on $Y$), then $A$ is surjective
    (onto $Y$).
  \item If $CA=I_X$ (the identity map on $X$), then $A$ is injective.
  \item If $AB=I_Y$ and $CA=I_X$, then $A$ is invertible, with
    $A^{-1}=B=C$.
  \item If $MN=NM$ and $N$ is invertible, then $MN^{-1}=N^{-1}M$.
  \end{enumerate}

  
\item[3.] Let $(X,\|\cdot\|)$ be a Banach
  space, and let $A\in\cL(X)$ be such that $\|A^p\|<1$ for some
  integer $p\geq 2$.  Show that $I-A\in\cL(X)$ is bijective, and that
  its inverse $(I-A)^{-1}:X\to X$ is also in $\cL(X)$.
  
\end{enumerate}


\end{document}