From Hahn-Banach to monotonicity / Stephen Simons.

Format:

Book

Author/Creator:

Simons, S.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1693.

Lecture notes in mathematics ; 1693

Language:

English

Subjects (All):

Monotone operators.

Monotonic functions.

Banach spaces.

Physical Description:

xiv, 244 pages : illustrations ; 24 cm.

Edition:

Second, expanded edition.

Place of Publication:

Berlin : Springer, 2008.

Summary:

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In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than using a "big convexification" of the graph of the multifunction and the "minimax technique" for proving the existence of linear functionals satisfying certain conditions, the Fitzpatrick function is used. The journey begins with a generalization of the Hahn-Banach theorem uniting classical functional analysis, minimax theory, Lagrange multiplier theory and convex analysis and culminates in a survey of current results on monotone multifunctions on a Banach space.

The first two chapters are aimed at students interested in the development of the basic theorems of functional analysis, which leads painlessly to the theory of minimax theorems, convex Lagrange multiplier theory and convex analysis. The remaining five chapters are useful for those who wish to learn about the current research on monotone multifunctions on (possibly non reflexive) Banach space.

Contents:

I The Hahn-Banach-Lagrange theorem and some consequences

1 The Hahn-Banach-Lagrange theorem 15

2 Applications to functional analysis 23

3 A minimax theorem 24

4 The dual and bidual of a normed space 25

5 Excess, duality gap, and minimax criteria for weak compactness 28

6 Sharp Lagrange multiplier and KKT results 32

II Fenchel duality

7 A sharp version of the Fenchel Duality theorem 41

8 Fenchel duality with respect to a bilinear form - locally convex spaces 44

9 Some properties of 1/2|.|2 49

10 The conjugate of a sum in the locally convex case 51

11 Fenchel duality vs the conjugate of a sum 54

12 The restricted biconjugate and Fenchel-Moreau points 58

13 Surrounding sets and the dom lemma 60

14 The θ-theorem 62

15 The Attouch-Brezis theorem 65

16 A bivariate Attouch-Brezis theorem 67

III Multifunctions, SSD spaces, monotonicity and Fitzpatrick functions

17 Multifunctions, monotonicity and maximality 71

18 Subdifferentials are maximally monotone 74

19 SSD spaces, q-positive sets and BC-functions 79

20 Maximally q-positive sets in SSD spaces 86

21 SSDB spaces 88

22 The SSD space E × E* 93

23 Fitzpatrick functions and fitzpatrifications 99

24 The maximal monotonicity of a sum 103

IV Monotone multifunctions on general Banach spaces

25 Monotone multifunctions with bounded range 107

26 A general local boundedness theorem 108

27 The six set theorem and the nine set theorem 108

28 D(S φ ) and various hulls 111

V Monotone multifunctions on reflexive Banach spaces

29 Criteria for maximality, and Rockafellar's surjectivity theorem 117

30 Surjectivity and an abstract Hammerstein theorem 123

31 The Brezis-Haraux condition 125

32 Bootstrapping the sum theorem 128

33 The > six set and the > nine set theorems for pairs of multifunctions 130

34 The Brezis-Crandall-Pazy condition 132

35 The norm-dual of the space E × E* and -functions 139

36 Subclasses of the maximally monotone multifunctions 147

37 First application of Theorem 35.8: type (D) implies type (FP) 153

38 T clb (E**), T clbn (B*) and type (ED) 154

39 Second application of Theorem 35.8: type (ED) implies type (FPV) 157

40 Final applications of Theorem 35.8: type (ED) implies strong 158

41 Strong maximality and coercivity 159

42 Type (ED) implies type (ANA) and type (BR) 161

43 The closure of the range 167

44 The sum problem and the closure of the domain 170

45 The biconjugate of a maximum and T clb (E**) 172

46 Maximally monotone multifunctions with convex graph 180

47 Possibly discontinuous positive linear operators 183

48 Subtler properties of subdifferentials 188

49 Saddle functions and type (ED) 192

VII The sum problem for general Banach spaces

50 Introductory comments 197

51 Voisei's theorem 197

52 Sums with normality maps 198

53 A theorem of Verona-Verona 199

VIII Open problems 203

IX Glossary of classes of multifunctions 205

X A selection of results 207.

Notes:

Includes bibliographical references and index.

ISBN:

9781402069185

1402069189

OCLC:

182663774