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 International Publishing

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

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).