CountableBasis 2002-01-07 10:11:59 2002-06-20 02:04:00 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A {\em countable basis} $\beta$ of a vector space $V$ over a field $F$ is a countable subset $\beta \subset V$ with the property that every element $v \in V$ can be written as an infinite series $$ v = \sum_{x \in \beta} a_x x $$ in exactly one way (where $a_x \in F$). We are implicitly assuming, without further comment, that the vector space $V$ has been given a topological structure or normed structure in which the above infinite sum is absolutely convergent (so that it converges to $v$ regardless of the order in which the terms are summed). The archetypical example of a countable basis is the Fourier series of a function: every continuous real-valued periodic function $f$ on the unit circle $S^1 = \mathbb{R}/2\pi$ can be written as a Fourier series $$ f(x) = \sum_{n=0}^\infty a_n \cos(n x) + \sum_{n=1}^\infty b_n \sin(n x) $$ in exactly one way. Note: A countable basis is a countable set, but it is not usually a basis.