\documentclass[12pt]{article}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{{\Large\bf Section 2.19} \bf 4, 6}
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}


\item[4.] Show that the convex hull of an open set $A$ in a topological vector
  space $X$ is open.
 \begin{quote}
    {\bf Note:}  Recall that $X$ being a topological vector
  space over the field $\KK$ requires that vector addition, $(x,y)\in
  X\times X\mapsto x+y\in X$, is continuous with respect to the product
  topology on $X\times X$; and that scalar multiplication,
  $(\alpha,x)\in\KK\times X\mapsto \alpha x\in X$  is continuous with respect to the product
  topology on $\KK\times X$.  Some implications of this are that, if
  $U,V\subset X$ are open, and $\alpha\in\KK$ is non-zero, then $U+V$
  and $\alpha U$ are also open.
  \end{quote}

    
\item[6.]
  \begin{enumerate}
    \item (Carath\'eodory's Theorem) Let $A\subset \RR^n$.  Show that any point $x\in \mathrm{co}(A)$
  can be expressed as a convex combination of at most $n+1$ points.

\item Using (a), show that the convex hull of a compact subset of
  $\RR^n$ is also compact.

\end{enumerate}

  
\end{enumerate}


\end{document}