Factorizing the Classical Inequalities (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

Bennett, Grahame

Softcover

ISBN 10: 0821804367 ISBN 13: 9780821804360

Publisher: American Mathematical Society, 1996

Synopsis

does not need NBB copy

About this title

Synopsis

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, $1^p\subseteq Y$, between certain (Banach) sequence spaces, the norm of the injection being the best constant of the particular inequality. The authors' approach is to replace $1^p$ by a larger space, $X$, with the properties: $\Vert 1^p\subseteq X\Vert =1$ and $\Vert X\subseteq Y\Vert =\Vert 1^p\subseteq Y\Vert$, 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, $\Vert 1^p\subseteq Y\Vert$, 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 space-alias 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, $Y=1^p\cdot Z$ (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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1996
• Language: English
• ISBN 10: 0821804367
• ISBN 13: 9780821804360
• Binding: Paperback