vector spaceVectorSpace22009-07-12 19:51:502009-07-12 19:51:50DefinitionIn standard LaTeX, \u and \v are accent commands (breve and caron), not mathematical vector variables. Inside math mode, make4ht/TeX4ht interprets them as accents and then produces the cascading brace errors you saw:
Missing } inserted.
Extra }, or forgotten \endgroup.\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
\renewcommand{\u}{\mathbf{u}}
\renewcommand{\v}{\mathbf{v}}
\newcommand{\w}{\mathbf{w}}
\newcommand{\0}{\mathbf{0}}Let $F$ be a field (or, more generally, a division ring). A \emph{vector space} $V$ over $F$ is a set with two operations, $+: V \times V \longrightarrow V$ and $\cdot: F \times V \longrightarrow V$, such that
\begin{enumerate}
\item $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in V$
\item $\u+\v=\v+\u$ for all $\u,\v\in V$
\item There exists an element $\0 \in V$ such that $\u+\0=\u$ for all $\u \in V$
\item For any $\u \in V$, there exists an element $\v \in V$ such that $\u+\v=\0$
\item $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in F$ and $\u \in V$
\item $1 \cdot \u = \u$ for all $\u \in V$
\item $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in F$ and $\u,\v \in V$
\item $(a+b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in F$ and $\u \in V$
\end{enumerate}
Equivalently, a vector space is a module $V$ over a ring $F$ which is a field (or, more generally, a division ring).
The elements of $V$ are called \emph{vectors}, and the element $\0 \in V$ is called the \emph{zero vector} of $V$.
This entry is a copy of the GNU FDL vector space article from \PMlinkexternal{PlanetMath}{http://planetmath.org/encyclopedia/VectorSpace.html}. Author of the original article: djao. History page of the original is \PMlinkexternal{here}{http://planetmath.org/?op=vbrowser&from=objects&id=364}