Contents

1 Preface vii
Chapter 1     The Normalized Duality Mapping

1.1  Definition and Some Fundamental Properties

1

1.2  Characterisations of Some Classes of Normed Spaces

4

1.3  Other Properties of Normalised Duality Mappings

10

Bibliography

17
Chapter 2     Semi-Inner Products in the Sense of Lumer-Giles

2.1 Definition and Fundamental Properties

19

2.2  Characterisation of Some Classes of Normed Spaces

21

2.3  Other Properties of L.-G.- s.i.p.s

25

Bibliography

29
Chapter 3  The Superior and Inferior Semi-Inner Products

3.1  Definition and Some Fundamental Properties

31

3.2  The Connection Between (. , . )s( i )  and the Duality Mapping

34

3.3  Other Properties of (. , . )s and (. , . )i

41

Bibliography

45
Chapter 4  Semi-Inner Products in the Sense of Milicic

4.1  Definition and the Main Properties

47

4.2  Normed Space of (G)-Type}

51

Bibliography

55
Chapter 5  (Q ) and (SQ) -Inner Product Spaces

5.1  (Q )- Inner Product Spaces

57

5.2  (SQ ) -Inner Product Spaces

63

Bibliography

69
Chapter 6  2k-Inner Products on Real Linear Spaces

6.1  Introduction

71

6.2  Main Properties of 2k-Inner Products

71

6.3  2k-Orthogonality

77

6.4  The Riesz Property

80

Bibliography

83
Chapter 7  Mappings Associated with the Norm Derivatives

7.1 Introduction

85

7.2  Some Mappings Associated with the Norm Derivatives

85

7.3  Properties of the Mapping dx,y

92

7.4  Properties of the Mapping gx,y

96

7.5  Properties of the (Fx,y)p Mappings

100

7.6  Properties of the Mappings (Yx,y)p

103

7.7  The Case of Inner Products

107

Bibliography

123
Chapter 8  Orthogonality in the Sense of Birkhoff-James

8.1  Definition and Preliminary Results

125

8.2  Characterisation of Some Classes of Normed Spaces

127

8.3  Birkhoff's Orthogonality and the Semi-inner Products

130

Bibliography

135
Chapter 9  Orthogonality Associated to the Semi-Inner Product

9.1  Orthogonality in the Sense of Giles

137

9.2  Orthogonality in the Sense of Milicic

141

9.3  The Superior and Inferior Orthogonality

145

Bibliography

149
Chapter 10  Characterisations of Certain Classes of Spaces

10.1  The Case of Giles Orthogonality

151

10.2  The Case of Milicic Orthogonality

157

Bibliography

161
Chapter 11  Orthogonal Decomposition Theorems

11.1  The Case of General Normed Linear Spaces

163

11.2  The Case of Smooth Normed Linear Spaces

164

11.3  The Case of (Q)-Banach and (SQ)-Banach Spaces

166

Bibliography

169
Chapter 12  Approximation of Continuous Linear Functionals

12.1  Introduction

171

12.2  A Characterisation of Reflexivity

171

12.3  Approximation of Continuous Linear Functionals

173 

12.4  A Characterization of Reflexivity in Terms of Convex Functions

176

Bibliography

179
Chapter 13  Some Classes of Continuous Linear Functionals

13.1  The Case of Semi-Inner Products

181

13.2  Some Classes of Functionals in Smooth Normed Spaces

184

13.3  Applications for Nonlinear Operators

186

13.4  The Case of General Real Spaces

189

13.5  Some Classes of Continuous Linear Functionals

190

13.6  Some Applications

193

Bibliography

195
Chapter 14  Smooth Normed Spaces of ( BD) -Type

14.1  Introduction

197

14.2  Smooth Normed Spaces of (D)-Type

197

14.3  Smooth Normed Spaces of (BD)-Type

201

14.4  Riesz Class of X*

206

14.5  Applications to Operator Equations

208

Bibliography

211
Chapter 15  Continuous Sublinear Functionals

15.1  Introduction

213

15.2  Semi-orthogonality in Reflexive Banach Spaces

213

15.3  Clins with the (H)-Property in Reflexive Spaces

216

15.4  Applications

220

Bibliography

223
Chapter 16  Convex Functions in Linear Spaces

16.1  Introduction

225

16.2  The Estimation of Convex Functions

226

16.3  Applications to Real Normed Linear Spaces

232

16.4  Applications in Hilbert Spaces

231

Bibliography

235
Chapter 17  Representation of Linear Forms

17.1  Introduction

237

17.2  Examples of Semi-Subinner Products

238

17.3  Representation of Linear Forms

239

17.4  Applications

242

Bibliography

245
A List of Papers on Semi-Inner Products 247
Index 255