The purpose of this book is to make available to the student some fundamentals of mathematical analysis. Specifically, it is intended to make such fundamentals available in a form that meets their need in many applications, like real analysis, integration, measure theory, and representation theory.
The principal point of view is to develop the basic structures of analysis, under which one can appropriately go on further in the domain of functional analysis. The book is intended to be essentially self-contained and accessible to advanced undergraduated students intended to Master degree courses. Its prerequisites are main standards from basics algebra and real analysis. In writing this book, we care about doing things as little abstract as possible. So, to make easy the access to the main concepts, each section of each chapter is illustrated by simple examples and exercises, which are mostly applications to concrete problems. References of treatises on the domain are given at the end. We hope that the book will reach the objectives assigned and especially will be useful to the teachers.
Dr. Lakhdar Meziani is presently a Professor of Mathematics at the Batna, University, Algeria. He spent more than 35 years in Teaching, Research and supervision of theses at various universities including University of Paris VI (France), University of Algiers, and King Abdulaziz University. He is graduated from University of Paris VI (France) and from Algiers University in the Öeld of Functional Analysis and Integral Representation Theory. He has published three books and many research papers in international journals of repute. His main research interests are Functional Analysis and Operators Theory.
Author
Lakhdar Meziani
Batna University, 05000 Batna
Algeria
Editor
Muhammad Imran
Government College University,
Faisalabad, Pakistan
Publication Date: 28 March 2021
Pages: 1-114
ISBN: updated soon
INTRODUCTION
- BASICS IN SET THEORY
1.1 Set Operations
1.2 Graphs, Binary Relations, Functions
1.3 Exercises
1.4 Binary Relations Properties
1.5 Exercises
1.6 The Real Number System
1.7 Exercises
1.8 Cardinals
1.9 Exercises - TOPOLOGICAL STRUCTURES
2.1 Topological Spaces
2.2 Exercises
2.3 Continuous Functions
2.4 Exercises
2.5 Separation Axioms
2.6 Exercises
2.7 Connected Spaces
2.8 Exercises - METRIC SPACES
3.1 Metrics
3.2 Exercises
3.3 Cauchy Sequences-Complete Spaces
3.4 Exercises
3.5 Uniformly Continuous Functions
3.6 Exercises
3.7 Countable Bases-Separable Spaces
3.8 Exercises
3.9 Baire Spaces
3.10 Exercises - COMPACT SPACES AND LOCALLY COMPACT SPACES
4.1 Compact Spaces
4.2 Exercises
4.3 Compact Metric Spaces
4.4 Exercises
4.5 Continuous Functions on Compact Spaces
4.6 Exercises
4.7 Product of Compact Spaces
4.8 Exercises
4.9 Locally Compact Spaces
4.10 Exercises
4.11 Compactification - BANACH SPACES
5.1 Normed spaces
5.2 Exercises
5.3 Linear Bounded Operators
5.4 Exercises
5.5 Normed Spaces of Finite Dimension
5.6 Exercises
5.7 Linear Bounded Operators in Banach Spaces
5.8 Exercises
5.9 Duality in Norm Spaces, Weak Topologies
5.10 Exercises - HILBERT SPACES
6.1 Hermitian Forms
6.2 Orthogonality
6.3 Orthonormal Bases
6.4 Dual Space
6.5 Exercises - TOPOLOGICAL VECTOR SPACES
7.1 Compatible Topology on a Vector Space
7.2 Complete Topological Vector Spaces - TOPOLOGICAL VECTOR SPACES OF FINITE DIMENSION
- THE SPACE C(X)
9.1 Stone-Weirstrass Theorem
9.2 Arzela -Ascoli Theorem - SEMI GROUPS OF LINEAR BOUNDED OPERATORS
10.1 C0 Semigroups
10.2 The Hille-Yosida Theorem
10.3 The Lumer Phillips Theorem
10.4 Complement: Uniformly Continuous Semigroups - MARKOV SEMI GROUPS AND TRANSITION FUNCTIONS
11.1 Transition Functions
11.2 Markovian Generators
11.3 Application - C0 SEMI GROUP OF CONTRACTIONS
- Bibliography
Articles
Books - Index
