The Dual of L∞(x, L,λ), Finitely Additive Measures and Weak Convergence: A Primer

By John Toland

In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X, L,λ)* with Lq(X, L,λ), where 1/p+1/q=1, as long as 1 L∞(X, L,λ)* 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,λ)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L∞(X, L,λ) 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∞(Rn) 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.

• Format: Paperback
• Language: English
• ISBN: 3030347311
• ISBN13: 9783030347314
• Release Date: February 2020
• Publisher: Springer
• Length: 99 Pages
• Weight: 0.37 lbs.
• Dimensions: 0.2" x 6.1" x 9.2"