This textbook describes selected topics in functional analysis as powerful tools of immediate use in many fields within applied mathematics, physics and engineering. It follows a very reader-friendly structure, with the presentation and the level of exposition especially tailored to those who need functional analysis but don’t have a strong background in this branch of mathematics. For every tool, this work emphasizes the motivation, the justification for the choices made, and the right way to employ the techniques. Proofs appear only when necessary for the safe use of the results. The book gently starts with a road map to guide reading. A subsequent chapter recalls definitions and notation for abstract spaces and some function spaces, while Chapter 3 enters dual spaces. Tools from Chapters 2 and 3 find use in Chapter 4, which introduces distributions. The Linear Functional Analysis basic triplet makes up Chapter 5, followed by Chapter 6, which introduces the concept of compactness. Chapter 7 brings a generalization of the concept of derivative for functions defined in normed spaces, while Chapter 8 discusses basic results about Hilbert spaces that are paramount to numerical approximations. The last chapter brings remarks to recent bibliographical items. Elementary examples included throughout the chapters foster understanding and self-study. By making key, complex topics more accessible, this book serves as a valuable resource for researchers, students, and practitioners alike that need to rely on solid functional analysis but don’t need to delve deep into the underlying theory.
Author(s): Carlos A. de Moura
Publisher: Springer
Year: 2022
Language: English
Pages: 222
City: Cham
Preface (to the Portuguese Version)
Conventions
Itinerary
Preface to the English Version
Contents
Notation
1 Road Map
1.1 Some Encounters
1.2 Feel Invited
2 Basic Concepts
2.1 Real Vector Spaces
2.2 Norm and Distance
2.3 Inner Product
2.4 Convergence
2.5 Continuous Functions
2.6 The Open, Closed, Dense Sets
2.7 The Cauchy Sequences
2.8 Quotient Spaces
2.9 Completion of a Normed Space
2.10 Principle of the Continuous Extension
2.11 The Linear Operators
2.12 Invertible Operators
2.13 Equivalent Norms
2.14 Lebesgue Integral
2.14.1 Introduction
2.14.2 Definition, Properties, and the Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript 1 Baseline left parenthesis double struck upper R right parenthesis) /StPNE pdfmark [/StBMC pdfmarkL1(R)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
2.14.3 Null Measure Sets
2.14.4 The Spaces ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript p Baseline left parenthesis double struck upper R right parenthesis comma 1 less than p less than normal infinity) /StPNE pdfmark [/StBMC pdfmark Lp(R) , 1 < p < ∞ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
2.14.5 The Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript normal infinity Baseline left parenthesis double struck upper R right parenthesis) /StPNE pdfmark [/StBMC pdfmark L∞(R)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
2.14.6 Convergence Theorems
2.14.7 Fubini Theorem and Differentiation Integration
2.14.8 Classical Bibliography Remarks
3 Dual of a Normed Space
3.1 Introduction
3.2 Linear Forms and Hyperplanes
3.3 Riesz Representation Theorem
3.3.1 Lax–Milgram Representation Theorem
3.3.2 An Application: Stokes Equation
3.4 The Projection Theorem
3.5 Representation for Some Dual Spaces
3.6 The Bidual Space
3.7 Radon-Nikodym Representation
3.8 Dirac Terminology
4 Sobolev Spaces and Distributions
4.1 Introduction and Notation
4.2 Sobolev Spaces Hk(Ω) and ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H 0 Superscript k Baseline left parenthesis upper Omega right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH0k(Ω)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
4.3 Weak Derivative and Regularization
4.4 The Distributions
4.5 Vector Functions and Distributions
4.6 The Trace Theorem
4.7 Sobolev Spaces of Real Order
4.7.1 δ-Function Representations
4.7.2 The Dual Space H-1(0,1)
4.7.3 The Spaces H-p, p Integer
4.7.4 The Spaces Hs, s Arbitrary Real
4.8 Fourier and Laplace Transforms
4.8.1 Fourier Transform for Functions
4.8.2 The Tempered Distributions
4.8.3 Laplace Transform
5 The Three Basic Principles
5.1 Introduction
5.2 Hahn-Banach Theorem
5.2.1 Application: A Dirichlet Problem
5.3 Open Mapping: Closed Graph
5.4 The Weak Convergence
5.5 The Uniform Boundedness Theorem
5.5.1 An Application to Numerical Schemes
6 Compactness
6.1 Introduction
6.2 Compactness in C0 and Lp
6.3 The Weak* Convergence
6.4 Rellich and Immersion Theorems
7 Function Derivatives in Normed Spaces
7.1 Introduction
7.2 Mean Value Theorems
7.3 Higher-Order Derivatives
7.4 Iterative Methods
8 Hilbert Bases and Approximations
8.1 Orthogonalization
8.2 Fourier Series
8.3 Separable Spaces: Approximation
8.3.1 An Example: The Finite Elements
8.4 Compactness: Eigenvectors Bases
8.5 Unbounded Operators
A Recent References
References
Index
