# Factorizing the Classical Inequalities

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*Grahame Bennett*

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 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, \(Y=
l^p\cdot Z\) (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.

#### Table of Contents

# Table of Contents

## Factorizing the Classical Inequalities

- Contents vii8 free
- §1 Introduction 110 free
- §2 Outline 514 free
- §3 The spaces d(a,p) and g(a,p) 817
- §4 Hardy 1322
- §5 Hölder 1928
- §6 Copson 2534
- §7 Two techniques 2938
- §8 Examples 3746
- §9 The meaning of l[sup(p)] 4251
- §10 ces(p) versus cop(p) 4756
- §11 Hilbert 5362
- §12 Köthe-Toeplitz duality 5766
- §13 The spaces l[sup(p)] d(a,q) 6473
- §14 Multipliers 6776
- §15 Some non-factorizations 7887
- §16 Examples 8493
- §17 Other matrices 9099
- §18 Summability matrices 97106
- §19 Hausdorff matrices 103112
- §20 Cesàro matrices 113122
- §21 Integral analogues 123132
- References 128137