Banach Limit and Applications

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Authors
1. Introduction
2. Hahn-Banach Theorem and Generalizations
	2.1. Extended form of Hahn-Banach Theorem
		2.1.1. Discussion
	2.2. Invariant linear functionals
	2.3. Hahn-Banach Theorem for semifields
	2.4. Sandwich version of Hahn-Banach Theorem
	2.5. Hahn-Banach Theorem for modules
3. Banach and Generalized Limits
	3.1. Banach limits
	3.2. Sublinear functionals generating Banach limits
	3.3. Sublinear functional that dominates all Banach limits
	3.4. Sublinear functionals involving regular matrices
	3.5. Invariant limits
	3.6. Generalizations of Banach limit
	3.7. Invariant means
	3.8. Further generalization
4. Almost Convergence
	4.1. Unique Banach limits
	4.2. Properties and examples of almost convergent sequences
		4.2.1. Examples of almost convergent sequences
		4.2.2. Periodic sequence
	4.3. Almost convergence and matrix method
	4.4. Almost convergence and uniform distribution of sequences
		4.4.1. Three parameter matrices
	4.5. Well-distributed sequences
	4.6. Almost convergence and statistical convergence
		4.6.1. Properties
	4.7. Almost convergence, density, and uniform density
	4.8. Almost convergence and uniform law of large numbers
	4.9. Almost convergence and almost ergodicity
5. Core Theorems
	5.1. Knopp\'s Core
	5.2. Banach Core
	5.3. Generalized CORE theorems
	5.4. Main theorems
6. Absolute Almost Convergence
	6.1. Introducing the new sequence space
	6.2. Emerging eight spaces
	6.3. Theorems
	6.4. Application of absolute almost convergence to Fourier series
	6.5. Tauberian Theorem
7. The Space ^l (p)
	7.1. New sequence Spaces ^l(p) and ^l(p)
		7.1.1. An important property
	7.2. Some topological properties
	7.3. Some further properties
	7.4. Some further topological properties
	7.5. Some outstanding conjectures
	7.6. Absolute Banach limits
8. Strong Almost Convergence
	8.1. Strong almost convergence
	8.2. Strong  ???? almost convergence
	8.3. On lacunary strong almost convergence
9. Functional Banach Limits
	9.1. Preliminaries
	9.2. Functional Banach limits
	9.3. The sets S and jSj
	9.4. Absolute almost convergence
	9.5. Strong almost convergence
10. Matrix Transformations
	10.1. Introduction
	10.2. The classes (`1(p); ^c) and (c(p)^c)
	10.3. The classes (`(p); ^c)
	10.4. The spaces ^c0(p) and ^c(p)
	10.5. The classes (v; ^c)
	10.6. The class (^c; c)
	10.7. The class (`1; ^ `p)
	10.8. The class (`1; ^ `p)
	10.9. The class (`p; ^ `p)
	10.10. Some open problems
11. Absolutely Almost and Strongly Almost Summable Sequences
	11.1. Introduction
	11.2. Absolute almost summability
	11.3. Strong almost summability
	11.4. Strong almost (Z; k) summability
	11.5. Spaces of strongly almost summable sequences
Bibliography
Index