Memoirs of the American Mathematical Society 1996; 130 pp; softcover Volume: 120 ISBN10: 0821804367 ISBN13: 9780821804360 List Price: US$43 Individual Members: US$25.80 Institutional Members: US$34.40 Order Code: MEMO/120/576
 This volume describes a new way of looking at the classical inequalities. The most famous such results (Hilbert, Hardy, and Copson) may be interpreted as inclusion relationships, \(l^p\subseteq Y\), between certain (Banach) sequence spaces, the norm of the injection being the best constant of the particular inequality. The authors' approach is to replace \(l^p\) by a larger space, \(X\), with the properties: \(\Vert l^p\subseteq X\Vert =1\) and \(\Vert X\subseteq Y\Vert =\Vert l^p\subseteq Y\Vert\), the norm on \(X\) being so designed that the former property is intuitive. Any such result constitutes an enhancement of the original inequality, because you now have the classical estimate, \(\Vert l^p\subseteq Y\Vert\), holding for a larger collection, \(X=Y\). The authors' analysis has some noteworthy features: The inequalities of Hilbert, Hardy, and Copson (and others) all share the same space \(Y\). That spacealias ces(\(p\) )being central to so many celebrated inequalities, the authors conclude, must surely be important. It is studied here in considerable detail. The renorming of \(Y\) is based upon a simple factorization, \(Y= l^p\cdot Z\) (coordinatewise products), wherein \(Z\) is described explicitly. That there is indeed a renorming, however, is not so simple. It is proved only after much preparation when duality theory is considered. Readership Graduate students and research mathematicians interested in real functions, functional analysis, and operator theory. Table of Contents  Introduction
 Outline
 The spaces \(d({\mathbf a}, p)\) and \(g({\mathbf a},p)\)
 Hardy
 Hölder
 Copson
 Two techniques
 Examples
 The meaning of \(\ell ^p\)
 \(ces(p)\) versus \(cop\) \((p)\)
 Hilbert
 KötheToeplitz duality
 The spaces \(\ell ^p\cdot d({\mathbf a},q)\)
 Multipliers
 Some nonfactorizations
 Examples
 Other matrices
 Summability matrices
 Hausdorff matrices
 Cesàro matrices
 Integral analogues
 References
