Convex Functional Analysis

by Kurdila, Andrew J.; Zabarankin, Michael

• ISBN13: 9783764321987
• ISBN10: 3764321989
• Format: Hardcover
• Copyright: 2005-08-30
• Publisher: Birkhauser

Summary

This volume is dedicated to the fundamentals of convex functional analysis. It presents those aspects of functional analysis that are extensively used in various applications to mechanics and control theory. The purpose of the text is essentially two-fold. On the one hand, a bare minimum of the theory required to understand the principles of functional, convex and set-valued analysis is presented. Numerous examples and diagrams provide as intuitive an explanation of the principles as possible. On the other hand, the volume is largely self-contained. Those with a background in graduate mathematics will find a concise summary of all main definitions and theorems.

Table of Contents

Preface	xi
1 Classical Abstract Spaces in Functional Analysis
1.1 Introduction and Notation	1	(4)
1.2 Topological Spaces	5	(16)
1.2.1 Convergence in Topological Spaces	13	(2)
1.2.2 Continuity of Functions on Topological Spaces	15	(2)
1.2.3 Weak Topology	17	(2)
1.2.4 Compactness of Sets in Topological Spaces	19	(2)
1.3 Metric Spaces	21	(20)
1.3.1 Convergence and Continuity in Metric Spaces	21	(2)
1.3.2 Closed and Dense Sets in Metric Spaces	23	(1)
1.3.3 Complete Metric Spaces	23	(2)
1.3.4 The Baire Category Theorem	25	(2)
1.3.5 Compactness of Sets in Metric Spaces	27	(3)
1.3.6 Equicontinuous Functions on Metric Spaces	30	(3)
1.3.7 The Arzela-Ascoli Theorem	33	(2)
1.3.8 Hölder's and Minkowski's Inequalities	35	(6)
1.4 Vector Spaces	41	(4)
1.5 Normed Vector Spaces	45	(7)
1.5.1 Basic Definitions	45	(1)
1.5.2 Examples of Normed Vector Spaces	46	(6)
1.6 Space of Lebesgue Measurable Functions	52	(6)
1.6.1 Introduction to Measure Theory	52	(2)
1.6.2 Lebesgue Integral	54	(3)
1.6.3 Measurable Functions	57	(1)
1.7 Hilbert Spaces	58	(7)
2 Linear Functionals and Linear Operators
2.1 Fundamental Theorems of Analysis	65	(10)
2.1.1 Hahn-Banach Theorem	65	(4)
2.1.2 Uniform Roundedness Theorem	69	(2)
2.1.3 The Open Mapping Theorem	71	(4)
2.2 Dual Spaces	75	(4)
2.3 The Weak Topology	79	(1)
2.4 The Weak* Topology	80	(8)
2.5 Signed Measures and Topology	88	(3)
2.6 Riesz's Representation Theorem	91	(4)
2.6.1 Space of Lebesgue Measurable Functions	91	(3)
2.6.2 Hilbert Spaces	94	(1)
2.7 Closed Operators on Hilbert Spaces	95	(2)
2.8 Adjoint Operators	97	(6)
2.9 Gelfand Triples	103	(3)
2.10 Bilinear Mappings	106	(5)
3 Common Function Spaces in Applications
3.1 The Lp Spaces	111	(2)
3.2 Sobolev Spaces	113	(13)
3.2.1 Distributional Derivatives	114	(3)
3.2.2 Sobolev Spaces, Integer Order	117	(1)
3.2.3 Sobolev Spaces, Fractional Order	118	(4)
3.2.4 Trace Theorems	122	(1)
3.2.5 The Poincaré Inequality	123	(3)
3.3 Banach Space Valued Functions	126	(11)
3.3.1 Bochner Integrals	126	(5)
3.3.2 The Space Lp((0,T),X)	131	(2)
3.3.3 The Space Wpq ((0,T),X)	133	(4)
4 Differential Calculus in Normed Vector Spaces
4.1 Differentiability of Functionals	137	(6)
4.1.1 Gateaux Differentiability	137	(2)
4.1.2 Fréchet Differentiability	139	(4)
4.2 Classical Examples of Differentiable Operators	143	(18)
5 Minimization of Punctionals
5.1 The Weierstrass Theorem	161	(2)
5.2 Elementary Calculus	163	(2)
5.3 Minimization of Differentiable Functionals	165	(1)
5.4 Equality Constrained Smooth Functionals	166	(5)
5.5 Fréchet Differentiable Implicit Functionals	171	(6)
6 Convex Functionals
6.1 Characterization of Convexity	177	(3)
6.2 Gateaux Differentiable Convex Functionals	180	(3)
6.3 Convex Programming in Rn	183	(5)
6.4 Ordered Vector Spaces	188	(5)
6.4.1 Positive Cones, Negative Cones, and Orderings	189	(2)
6.4.2 Orderings on Sobolev Spaces	191	(2)
6.5 Convex Programming in Ordered Vector Spaces	193	(6)
6.6 Gateaux Differentiable Functionals on Ordered Vector Spaces	199	(6)
7 Lower Semicontinuous Punctionals
7.1 Characterization of Lower Semicontinuity	205	(3)
7.2 Lower Semicontinuous Functionals and Convexity	208	(4)
7.2.1 Banach Theorem for Lower Semicontinuous Functionals	208	(2)
7.2.2 Gateaux Differentiability	210	(1)
7.2.3 Lower Semicontinuity in Weak Topologies	210	(2)
7.3 The Generalized Weierstrass Theorem	212	(9)
7.3.1 Compactness in Weak Topologies	213	(2)
7.3.2 Bounded Constraint Sets	215	(1)
7.3.3 Unbounded Constraint Sets	215	(2)
7.3.4 Constraint Sets on Ordered Vector Spaces	217	(4)
References	221	(2)
Index	223

Excerpts

"This volume is dedicated to the fundamentals of convex functional analysis. It presents those aspects of functional analysis that are extensively used in various applications to mechanics and control theory." "The purpose of the text is essentially two-fold. On the one hand, a bare minimum of the theory required to understand the principles of functional, convex and set-valued analysis is presented. Numerous examples and diagrams provide as intuitive an explanation of the principles as possible. On the other hand, the volume is largely self-contained. Those with a background in graduate mathematics will find a concise summary of all main definitions and theorems."--BOOK JACKET.