Semi-Inner Products and Applications

1. Preface......Page 7
1. Definition and Some Fundamental Properties......Page 9
2. Characterisations of Some Classes of Normed Spaces......Page 12
3. Other Properties of Normalised Duality Mappings......Page 18
Bibliography......Page 23
1. Definition and Fundamental Properties......Page 27
2. Characterisation of Some Classes of Normed Spaces......Page 29
3. Other Properties of L-G-s.i.p.s......Page 33
Bibliography......Page 34
1. Definition and Some Fundamental Properties......Page 39
2. The Connection Between ( ,) s( i)  and the Duality Mapping......Page 42
3. Other Properties of ( ,) s and ( ,) i......Page 49
Bibliography......Page 52
1. Definition and the Main Properties......Page 55
2. Normed Space of ( G) -Type......Page 59
Bibliography......Page 60
1. ( Q)  -- Inner Product Spaces......Page 65
2. ( SQ) - Inner Product Spaces......Page 71
Bibliography......Page 75
2. Main Properties of 2k-Inner Products......Page 79
3. 2k-Orthogonality......Page 85
4. The Riesz Property......Page 88
Bibliography......Page 89
2. Some Mappings Associated with the Norm Derivatives......Page 93
3. Properties of the Mapping 0=x\"010Ex,y......Page 100
4. Properties of the Mapping 0=x\"010Dx,y......Page 104
5. Properties of x,yp Mappings......Page 108
6. Properties of the Mappings x,yp......Page 111
7. The Case of Inner Products......Page 115
Bibliography......Page 130
1. Definition and Preliminary Results......Page 133
2. Characterisation of Some Classes of Normed Spaces......Page 135
3. Birkhoff\'s Orthogonality and the Semi-inner Products......Page 138
Bibliography......Page 142
1. Orthogonality in the Sense of Giles......Page 145
2. Orthogonality in the Sense of Milicic......Page 149
3. The Superior and Inferior Orthogonality......Page 153
Bibliography......Page 156
1. The Case of Giles Orthogonality......Page 159
2. The Case of Milicic Orthogonality......Page 165
Bibliography......Page 167
1. The Case of General Normed Linear Spaces......Page 171
2. The Case of Smooth Normed Linear Spaces......Page 172
3. The Case of ( Q) -Banach and ( SQ) -Banach Spaces......Page 174
Bibliography......Page 175
2. A Characterisation of Reflexivity......Page 179
3. Approximation of Continuous Linear Functionals......Page 181
4. A Characterization of Reflexivity in Terms of Convex Functions......Page 184
Bibliography......Page 185
1. The Case of Semi-Inner Products......Page 189
2. Some Classes of Functionals in Smooth Normed Spaces......Page 192
3. Applications for Nonlinear Operators......Page 194
4. The Case of General Real Spaces......Page 197
5. Some Classes of Continuous Linear Functionals......Page 198
Bibliography......Page 201
2. Smooth Normed Spaces of ( D) -Type......Page 205
3. Smooth Normed Spaces of ( BD) -Type......Page 209
4. Riesz Class of X......Page 214
5. Applications to Operator Equations......Page 216
Bibliography......Page 218
2. Semi-orthogonality in Reflexive Banach Spaces......Page 221
3. Clins with the ( H) -Property in Reflexive Spaces......Page 224
4. Applications......Page 228
Bibliography......Page 229
1. Introduction......Page 233
2. The Estimation of Convex Functions......Page 234
3. Applications to Real Normed Linear Spaces......Page 239
4. Applications in Hilbert Spaces......Page 240
Bibliography......Page 242
1. Introduction......Page 245
2. Examples of Semi-Subinner Products......Page 246
3. Representation of Linear Forms......Page 247
4. Applications......Page 250
Bibliography......Page 252
A List of Papers on Semi-Inner Products......Page 253
Index......Page 262