Geometry of Orlicz spaces

Chen Shutao

GDML_Books, (1996), p.

Abstract

CONTENTSPreface..............................................................................................................................4Introduction........................................................................................................................51. Orlicz spaces..................................................................................................................6 1.1. Orlicz functions...........................................................................................................6 1.2. Orlicz spaces............................................................................................................13 1.3. Orlicz norm...............................................................................................................17 1.4. Luxemburg norm......................................................................................................23 1.5. Bounded linear functionals.......................................................................................27 1.6. Weak topology.........................................................................................................32 1.7. Norm attainable functionals......................................................................................38 1.8. Isomorphic subspaces..............................................................................................41 1.9. Basis........................................................................................................................472. Convexity and smoothness..........................................................................................51 2.1. Extreme points and rotundity....................................................................................51 2.2. λ property.................................................................................................................58 2.3. Locally uniform rotundity..........................................................................................66 2.4. Mid-point locally uniform rotundity and uniform rotundity in every direction.............74 2.5. Uniform k-rotundity...................................................................................................84 2.6. Full convexity and weakly uniform rotundity.............................................................90 2.7. Smoothness.............................................................................................................953. Other geometrical properties......................................................................................107 3.1. Normal structure.....................................................................................................107 3.2. H-property..............................................................................................................119 3.3. Nonsquareness......................................................................................................127 3.4. Some geometrical characterization of reflexivity.....................................................135 3.5. Roughness, girth and Radon-Nikodym property.....................................................139 3.6. Ball-packing constants...........................................................................................1454. Some applications of geometry of Orlicz spaces........................................................156 4.1. Best approximation.................................................................................................156 4.2. Predictors...............................................................................................................160 4.3. Some optimal control problems..............................................................................1655. Geometry of Musielak-Orlicz spaces..........................................................................175 5.1. Musielak-Orlicz spaces...........................................................................................175 5.2. Extreme points, rotundity and uniform rotundity......................................................179 5.3. Complex rotundities................................................................................................187Bibliography...................................................................................................................192Index..............................................................................................................................203

Preface60 years ago, in 1932, there appeared both the famous book on functional analysis by S. Banach, Théorie des opérations linéaires, and the article on spaces, later called Orlicz spaces, by W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B in Bull. Internat. Acad. Polon. Sci. Sér. A. The latter notion was an important extension of the notion of $L_p$ and $l_p$ spaces, introduced by F. Riesz in 1910 and 1913, respectively. The investigations of geometric properties of Banach spaces, i.e., properties which are invariant with respect to linear isometries, date back to 1936, when J. A. Clarkson introduced the notion of uniformly rotund spaces in the paper Uniformly convex spaces in Trans. Amer. Math. Soc. 40, and it was shown that $L_p$ with 1 < p < ∞ are examples of such spaces. Between the two notions of uniform rotundity and rotundity of a Banach space, a number of intermediate geometric properties have recently been investigated. Applications were found in such seemingly distant branches of mathematics as approximation theory and probability theory. Now, the scale of $L_p$ spaces seems to be too narrow in order to provide a good model for distinguishing subtleties connected with various geometric properties of Banach spaces. A much richer field of examples is obtained by considering Orlicz spaces $L_M$ of functions and $l_M$ of sequences, where M is an Orlicz function. Also, one distinguishes in Orlicz spaces two norms, the Orlicz norm ⃦· ⃦° and the Luxemburg norm ⃦· ⃦, which are equivalent, but the identity operator from ($L_M$, ⃦· ⃦°) to ($L_M$, ⃦· ⃦) is not a linear isometry, which implies that from the point of view of geometric properties, these spaces differ essential.The importance of this book lies in the fact that it is the first book in English devoted to the problem of geometric properties of Orlicz spaces, and that it provides complete, up-to-date information in this domain. In most cases the theorems concern necessary and sufficient conditions for a given geometric property expressed by properties of the function M which generates the space $L_M$ or $l_M$. Some applications to best approximation, predictors and optimal control problems are also discussed.This book shows the great role played recently by the Harbin School of Functional Analysis in problems of geometric properties of Orlicz spaces. There are many results in this book which have so far been published only in Chinese.Anyone interested in the domain of geometric properties of Banach spaces will certainly find the present book indispensable.

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Zbl 1089.46500

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