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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{\bf {\Large Section 2.5:} 1, 2 (optional), 3 (optional), 4}
\end{center}
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\begin{enumerate}
  
\item[1.]
  \begin{enumerate}
    \item Given any $x\in\KK^n$, show that
      $\|x\|_\infty=\lim_{p\to\infty}\|x\|_p$.
    \item Given any $x\in\ell^{\infty}$, show that
      \begin{align*}
        \|x\|_\infty=\lim_{n\to\infty}\left(\lim_{p\to\infty}\left(\sum_{k=1}^n|x_k|^p\right)^{1/p}\right)
      \end{align*}
  \end{enumerate}
\begin{quote}
{\bf Hint:}  For the first part, argue that $\|x\|_\infty\leq
\|x\|_p\leq n^{1/p}\|x\|_\infty$.
\end{quote}

\item[2.] \textbf{(Optional)} The exercise concerns the conditions under
  which equality is achieved in H\"older's Inequality and Minkowski's Inequality.
  \begin{enumerate}
  \item Let $p,q> 1$ be conjugate exponents. Show that equality in
    H\"older's Inequality,
    \begin{align}\label{Holder}
      \sum_{k=1}^\infty |x_ky_k|\leq \|x\|_p\|y\|_q\mbox{ for }x\in\ell^p\,,\,y\in\ell^q~,
    \end{align}
    holds if and
      only if there are $a,b\geq 0$, at least one of
      which is positive, such that $a|x_k|^p=b|y_k|^q$ for all
      $k\in\NN$.
     
    \item Let $p> 1$.  Show that equality holds in Minkowski's
      Inequality
      \begin{align}\label{Minkowski}
        \|x+y\|_p\leq \|x\|_p+\|y\|_p\mbox{ for }x,y\in\ell^p~,
      \end{align}
      if and only if there are $a,b\geq 0$, at least one of which is
      positive, for which $a x_k=b y_k$ for all $k\in\NN$.
  \end{enumerate}


\item[3.] \textbf{(Optional)} Let $0<p<1$, and
  $X=\left\{x\in\KK^\NN:\,\sum_{k=1}^\infty|x_k|^p<\infty\right\}$.
  \begin{enumerate}
  \item Show that $X$ is a vector space.
 
   
  \item Show that the mapping $x\mapsto \rho(x)\doteq
    \left(\sum_{k=1}^\infty|x_k|^p\right)^{1/p}$ is not a norm on $X$.

    \item Define $d:X\times X\to\RR$ by
      $d(x,y)=\sum_{k=1}^\infty|x_k-y_k|^p$ for $x,y\in X$.  Show that
      $d$ is a metric on $X$.
  
  \end{enumerate}

  
\item[4.]   Let $p$ and $q$ satisfy $0<p<q<\infty$ (do not assume
  $p\geq 1$), and let $x\in\KK^\NN$ satisfy $\sum_{k=1}^\infty
  |x_k|^p<\infty$.  Show that the series $\sum_{k=1}^\infty
  |x_k|^q$ converges, and that Jensen's inequality holds:
  \begin{align}\label{JensenIneq}
    \left(\sum_{k=1}^\infty |x_k|^q\right)^{1/q}\leq \left(\sum_{k=1}^\infty |x_k|^p\right)^{1/p}~.
  \end{align}
  \begin{quote} {\bf Note:} There are a variety of related and
    important inequalities that go by the name ``Jensen's
    inequality''.  Although you might find a version that can be
    properly applied to yield~\eqref{JensenIneq}, it is probably
    simpler to just prove~\eqref{JensenIneq} directly.
  \end{quote}
    \begin{quote} {\bf Hint:} The case $x=0$ is trivial, so note this
      in your argument, and quickly move onto the case $x\neq 0$.
      There will be an $N$ such that, for $n\geq N$,
      $\sum_{k=1}^n|x_k|^q>0$.
      Fix $n\geq N$, and consider $\sum_{k=1}^n|y_k|^q$ and
      $\sum_{k=1}^n|y_k|^p$, where
      $y_k=|x_k|/(\sum_{k=1}^n|x_k|^q)^{1/q}$.  How do these two sums
      compare with each other?
  \end{quote}
\end{enumerate}


\end{document}