Completion of a normed space
= Completion of a normed space
The completion of a <normed vector space> $X$ is a <Banach space> containing an isometric dense copy of $X$. Construct it from <Cauchy sequences> modulo <sequences> whose difference tends to zero; define the <norm> by $\|[(x_n)]\|=\lim_n\|x_n\|$. Equivalently, the canonical evaluation map into the <bidual space> is an isometry by the <Hahn-Banach theorem>, and the closure of its image supplies a completion.