Complete 2001-10-27 18:24:47 2008-01-23 16:08:38 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A metric space $X$ is {\em complete} if every \PMlinkname{Cauchy sequence}{CauchySequence} in $X$ is a convergent sequence. {\bf Examples:} \PMlinkescapephrase{Cauchy sequence} \begin{itemize} \item The space $\mathbb{Q}$ of rational numbers is not complete: the sequence $3$, $3.1$, $3.14$, $3.141$, $3.1415$, $3.14159$, $3.141592\ldots$ consisting of finite decimals converging to $\pi \in \mathbb{R}$ is a Cauchy sequence in $\mathbb{Q}$ that does not converge in $\mathbb{Q}$. \item The space $\mathbb{R}$ of real numbers is complete, as it is the completion of $\mathbb{Q}$ with respect to the standard metric (other completions, such as the $p$-adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space. \item Every Banach space is complete. For example, the $L^p$--space of p-integrable functions is a complete metric space if $p \geq 1$. \end{itemize}