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\begin{document}

\title{Functional Analysis}
\author{Homework 3}
\date{due March 4, 2009}
\maketitle

\begin{enumerate}

\item Let $X$ be a normed vector space and $C\subset X$ an open,
  convex subset containing the origin.  Define $p \colon X \to
  [0,\infty)$ via
\[
  p(x) = \inf \{ \alpha>0 \colon x/\alpha \in C \} \,.
\]
Prove that
\begin{enumerate}
\item $p$ is sublinear;
\item $p$ is bounded, i.e.\ there exists $m>0$ such that for every
  $x\in X$,
\[
  0 \leq p(x) \leq m \, \lVert x \rVert \,;
\]
\item $C = \{ x \in X \colon p(x)<1 \}$.

\end{enumerate}


\item Folland, p.\ 164--165, no.\ 27, 34, 35, 37, 38.


\end{enumerate}

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