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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\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 }}}
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\newcommand{\enorm}[1]{\ensuremath{|\!|\!| #1|\!|\!| }}

\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator*{\supp}{supp}
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\DeclareMathOperator*{\Range}{Im}
\DeclareMathOperator*{\spn}{span}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{{\Large\bf Section 2.12: }\bf A, B, 6; Optional: C, 1, 5}
\end{center}

\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}

\item[A.] Let $A:X\to Y$ be a linear operator between these vector
  spaces.
  \begin{enumerate}
  \item Show that the kernel $\Ker(A)$ and direct image $\Range(A)$ 
    are subspaces of their respective vector spaces.
    
  \item Show that, if $X$ and $Y$ are normed vector spaces and $A$ is
    continuous, then $\Ker(A)$ is closed.
    
  \item Let $X=C^1[0,1]$ with the sup-norm
    $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and let $Y=\RR$ with the absolute
    value norm.  Let $A:X\to Y$ be given by $A f\doteq f'(1)$.  Show
    that $A$ is a linear operator but that $\Ker(A)$ is not closed.
    Hint: Consider the sequence of polynomials $p_n(x)=x-x^n/n$ for $n\in\NN$.
  \end{enumerate}

\item[B.] Let $X$ be a vector space over $\KK$ and let $\phi:X\to\KK$
  be a linear functional.
  \begin{enumerate}
    \item Let $U$ be a subspace of $X$.  Show that if
      $\Ker(\phi)\subset U$, then $\Ker(\phi)=U$ or $U=X$.
    \item Let $\psi:X\to\KK$ also be a linear functional.  If
        $\Ker(\phi)\subset\Ker(\psi)$, then there is a $\beta\in\KK$
        such that $\psi=\beta \phi$.
    \item If $\Ker(\phi)\neq X$ and $z\in X\setminus\Ker(\phi)$,
        then $X=\Ker(\phi)\oplus\spn\{z\}$.
  \end{enumerate}
  Hint:  You will get a lot of mileage out of constructions of the
  form $x=x-\alpha z+\alpha z$, where $\alpha=\phi(x)/\phi(z)$.
  
\item[C.]\textbf{(Optional)} Show that the space $\cL(X,Y)$ of bounded linear
  operators from $X$ to $Y$ is a vector space, under the usual
  definitions of operator/function addition and scalar
  multiplication.  Don't verify all of the properties of that must
  hold for a vector space, but only show that $\cL(X,Y)$ is closed
  under addition and scalar multiplication.

\item[1.]\textbf{(Optional)} Let $\|\cdot\|_p$ for $1\leq p\leq \infty$ be the norms on
  $\KK^n$ defined by in Theorem 2.2.2, and identify a matrix
  $A\in\KK^{n\times n}$ with a linear operator $F\in \cL(\KK^n)$ in
  the usual way, i.e.
  \begin{align*}
    A=
    \begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\
      \vdots&\vdots&&\vdots\\
      a_{n1}&a_{n2}&\cdots&a_{nn}
    \end{pmatrix}
    \leftrightarrow 
    F(x)=Ax=\begin{pmatrix}\sum_{j=1}^na_{1j}x_j\\\sum_{j=1}^na_{2j}x_j\\\vdots\\
      \sum_{j=1}^na_{nj}x_j
    \end{pmatrix}
  \end{align*}
  \begin{enumerate}
  \item Show that
    \begin{align*}
      \|A\|_1&\doteq\sup_{x\neq 0}\frac{\|Ax\|_1}{\|x\|_1}=\max_{1\leq
               j\leq n}\sum_{i=1}^n|a_{ij}|~,\\
     \|A\|_\infty&\doteq\sup_{x\neq 0}\frac{\|Ax\|_\infty}{\|x\|_\infty}=\max_{1\leq
                   i\leq n}\sum_{j=1}^n|a_{ij}|~,\\
     \|A\|_2&\doteq\sup_{x\neq
              0}\frac{\|Ax\|_2}{\|x\|_2}=\max\{|\lambda|^{1/2}:\,\lambda\mbox{
              is an eigenvalue of } A^*A\}~,
    \end{align*}
     where $A^*$ is the conjugate transpose of $A$ when $A$ is complex,
    and just the transpose of $A$ when $A$ is real.
    
  \item Find formulas in the spirit of part (a) for the operator norms
    defined by
    \begin{align*}
      \|A\|_{p,1}\doteq \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_1}\;
      (1<p\leq \infty)\quad,\quad \|A\|_{\infty,p}\doteq \sup_{x\neq 0}\frac{\|Ax\|_\infty}{\|x\|_p}\;
      (1\leq p< \infty)~.
    \end{align*}
 \end{enumerate}


 \item[5.] \textbf{(Optional)} Show that a normed vector space $(X,\|\cdot\|)$ is finite dimensional
   if and only if all linear functionals on $X$ are continuous.
 
 
  \item[6.] Let $X$ and $Y$ be two normed vector spaces over the same
    field $\KK$, and let $A:X\to Y$ be a linear operator.
    \begin{enumerate}
    \item We define $[A]: X/\Ker A\to\Range A$ by $[A][x]\doteq Ax$
      for each $[x]\in X/\Ker A$.  Show that this definition makes
      sense.  At issue here is the fact that, if $\tilde{x}\in[x]$,
      then $[x]=[\tilde{x}]$, so we need $[A][\tilde{x}]=[A][x]$ if
      the definition is to make sense.
  
    \item Show that, if $A$ is continuous, then $[A]$ is continuous
      and $\|[A]\|=\|A\|$.
    \end{enumerate}
    
 



  
\end{enumerate}


\end{document}