Lectures on Functional Analysis and the Lebesgue Integral

1. Front Matter

   Pages i-xx

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2. Functional Analysis

   1. Front Matter

      Pages 1-2

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   2. Hilbert Spaces

      • Vilmos Komornik

      Pages 3-54

   3. Banach Spaces

      • Vilmos Komornik

      Pages 55-117

   4. Locally Convex Spaces

      • Vilmos Komornik

      Pages 119-147

3. The Lebesgue Integral

   1. Front Matter

      Pages 149-149

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   2. Monotone Functions

      • Vilmos Komornik

      Pages 151-167

   3. The Lebesgue Integral in \(\mathbb{R}\)

      • Vilmos Komornik

      Pages 169-195

   4. Generalized Newton–Leibniz Formula

      • Vilmos Komornik

      Pages 197-209

   5. Integrals on Measure Spaces

      • Vilmos Komornik

      Pages 211-254

4. Function Spaces

   1. Front Matter

      Pages 255-256

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   2. Spaces of Continuous Functions

      • Vilmos Komornik

      Pages 257-304

   3. Spaces of Integrable Functions

      • Vilmos Komornik

      Pages 305-340

   4. Almost Everywhere Convergence

      • Vilmos Komornik

      Pages 341-362

5. Back Matter

   Pages 363-403

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This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small ℓ^p spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým.

Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erdős and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they arecombined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included.

Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field.

• Functional analysis
• Hilbert space
• normed space
• Banach space
• dual space
• weak convergence
• reflexivity
• locally convex space
• Lebesgue integral
• spaces of continuous functions
• spaces of integrable functions

“This book is written in a clear and readable style … . The author gathers a collection of exercises in each chapter and presents some hints and solutions to some of them at the end of the book, helping the readers to develop their knowledge. The book is indeed a comprehensive study of L^p-spaces, useful for graduate students in mathematics, physics and engineering.” (Mohammad Sal Moslehian, zbMATH 1350.46002, 2017)

• University of Strasbourg, Strasbourg, France

  Vilmos Komornik

Vilmos Komornik has studied in Budapest, Hungary, and has taught in Hungary and France for nearly 40 years. His main research fields are control theory of partial differential equations and combinatorial number theory. He has made a number of contributions to the theory of J.L. Lions on exact controllability and stabilization and has co-authored several papers on expansions in noninteger bases with P. Erdős.

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