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

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

by John Toland
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Book cover type: Paperback
  • ISBN13: 9783030347314
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 99
  • Original Price: EUR 64.99
  • Language: English
  • Edition: 2020
  • Item Weight: 181 grams
  • BISAC Subject(s): Mathematical Analysis, Functional Analysis, and Calculus

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.

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).

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