\documentclass{article} \usepackage{amstex,theorem} \textwidth 5.6in \topmargin=-.5in \textheight=8.0in \oddsidemargin=0.5in \evensidemargin=0.5in \textwidth=5.5in \begin{document} \title{Computational scales of Sobolev norms } \author{\underline{James H. Bramble}, Joseph E. Pasciak and Panayot S. Vassilevski} \maketitle \begin{abstract} In this talk, abstract results concerning representations of norms on scales of Sobolev spaces will be presented. These representations are defined in terms of a sum of differences of linear operators which map into neighboring multilevel spaces. We apply these results to define efficient computational Sobolev norms for $H^s(\Omega)$ when $-3/2