Linear Functionals

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN: 978-1-0677109-9-6

Summary

The word functional is a noun, meaning a function whose argument is a function. Functional analysis mainly studies operators and functionals on infinite dimensional spaces. It summarizes the results of classical analysis and function theories; composes the algebraic and topological structure of a space into a purely abstract form. This book summarizes basic concepts and major theorems for linear functionals, include the four pillars of functional analysis. It can be used for the study of differential and integral equations, quantum fields, function approximation, numerical analysis, mathematical programming, variation, etc.

 

Chapter 1 Concept of Space          4

• 1 Metric Space 4

fixed-point theorems      5

Completion of a Metric Space       8

Complete Function Spaces on a General Compact Set        10

 

• 2 Normed Vector Spaces 15

Vector Spaces        15

Semi-norms and norms        18

Banach Spaces, Sobolev space       19

Sublinear Functionals     23

Schauder’s Fixed-Point Theorem      24

 

• 3 Inner Product Space 25

Hilbert Spaces      

Inner Products induces norms and distances

Orthogonality and orthogonal basis

Applications: Best Approximation, least square method, splines

Hilbert dimension

 

• 4 Separable Spaces 39

Direct sums

Tensor products

 

Chapter 2 Linear Operators and Functionals           41

• 1 Linear operators and Dual space 41
• 2 Riesz Representation Theorem 43

Solution of the Laplace equation     44

• 3 Properties of Linear Operator 46

Baire’s Category Theorem          47

The Open Mapping Theorem         49

The Closed Graph Theorem        52

Uniform boundedness Theorem        53

Closed Operators       56

 

• 4 Hahn-Banach Theorem 59

Extensions of Linear Functionals

Hahn–Banach theorem (for X over real number field)         60

Hahn–Banach Theorem (Over complex field)         61

The Geometric Form of Hahn Banach Theorem            63

 

Chapter 3 Conjugate and Reflexive Spaces     66

• 1 Adjoin Operators 66

Unbounded Operators        67

Reflexivity        68

• 2 Spectra of linear Operators 71

Projections, Complements and Reductions       79

The Ascent and Descent of an Operator        82

 

Chapter 4 Function Space L^p

• 1 The Complete Space L^p 85
• 2 The Hilbert Space L² 88

 

Chapter 5 Fourier Analysis         93

• 1 Convergence of Fourier Series 93
• 2 C-1 Summability of Fourier Series 95
• 3 Fourier Transform on L¹(R¹) 96
• 4 Fourier Transform on L²(R¹) 100

 

Chapter 6 Generalized Functions         102

• 1 Concept 102
• 2 Properties 106
• 3 Operations on Generalized Functions 108