\subsection{ Mapping of Function Spaces  to a Finite Number of Dimensions}



\begin{itemize}
\item In solving differential equations numerically, we take infinite
dimensional operators on space of functions and convert them to
finite matrix operators in space of $n$ dimensional vectors.
\vspace{12pt}

\item The different numerical approaches correspond to different ways
of mapping the infinite dimensional space to a finite space---there is
no unique way and some are much better than others.
\end{itemize}