The Dual of L¿(X,L,¿), Finitely Additive Measures and Weak Convergence

A Primer
1. Auflage 2020
ISBN: 978-3-030-34731-4
Verlag: Springer

Buch, Englisch, 99 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 184 g

Reihe: SpringerBriefs in Mathematics

In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space L (X,L,?)* with L (X,L,?), where 1/p+1/q=1, as long as 1 = p<8. However, L (X,L,?)* cannot be similarly described, and is instead represented as a class of finitely additive measures.

This book provides a reasonably elementary account of the representation theory of L (X,L,?)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L (X,L,?) to be weakly convergent, applicable in the one-point compactification of X, is given.

With a clear summary of prerequisites, and illustrated by examples including L ( R ) and the sequence space l , this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.

Zielgruppe

Research

Autoren/Hrsg.

Toland, John

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1 Introduction.- 2 Notation and Preliminaries.- 3  L  and its Dual.- 4 Finitely Additive Measures.- 5 G: 0-1 Finitely Additive Measures.- 6 Integration and Finitely Additive Measures.- 7 Topology on G.- 8 Weak Convergence in  L (X,L,?).- 9  L * when X is a Topological Space.- 10 Reconciling Representations.- References.- Index.

Über Autor(innen)

John Toland FRS is a mathematical analyst who worked in nonlinear partial differential equations and served as Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (2011-2016). He was awarded the London Mathematical Society Berwick Prize (2000) and the Royal Society Sylvester Medal (2012).