Hans,

A second example for today. This file shows the concern that members of the
AMS have for spacing around equal signs in aligned environments. Contrast
the align with the standard eqnarray from latex2e.

\documentclass{article}
\usepackage{amsmath}
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\eps}{\varepsilon}
\DeclareMathOperator{\RE}{Re} \DeclareMathOperator{\IM}{Im}\begin{document}
\newcommand{\seq}[1]{\left<#1\right>}

\begin{align*}
  \norm{f_0 - (I+\eps A)f}^2
  &= \norm{f_0-f - \eps A f}^2 \\
  &= \norm{f_0-f}^2 - 2\eps\RE\seq{A{f},f_0-f} + \eps^2\norm{A{f}}^2 \\
  &< \norm{f_0-f}^2 - 2\eps\delta + \eps^2 4\norm{A}^2 \\
  &= \norm{f_0-f}^2 - 2\eps(\delta - \eps 2\norm{A}^2) \\
  &< \norm{f_0-f}^2.
\end{align*}

\begin{eqnarray*}
  \norm{f_0 - (I+\eps A)f}^2
  &=& \norm{f_0-f - \eps A f}^2 \\
  &=& \norm{f_0-f}^2 - 2\eps\RE\seq{A{f},f_0-f} + \eps^2\norm{A{f}}^2 \\
  &<& \norm{f_0-f}^2 - 2\eps\delta + \eps^2 4\norm{A}^2 \\
  &=& \norm{f_0-f}^2 - 2\eps(\delta - \eps 2\norm{A}^2) \\
  &<& \norm{f_0-f}^2.
\end{eqnarray*}

\end{document}