Semi-Inner Products and Applications

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 d_x,y	92
7.4 Properties of the Mapping g_x,y	96
7.5 Properties of the (F_x,y)^p Mappings	100
7.6 Properties of the Mappings (Y_x,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