Ⅰ BANACH SPACES 3
1．On structure 3
2．The axioms 4
3．Linear functionals 10
4．The canonical map 15
5．Subspaces and orthogonality 16
6．The Hahn－Banach theorem 18
7．Other topologies 23
8．Examples and exercises 28
Ⅱ LINEAR TRANSFORMATIONS 32
1．Preliminaries 32
2．The adjoint transformation 36
3．The boundedness of the inverse transformation 37
4．Closed transformations 42
5．The uniform boundedness principle 45
6．Projections 46
7．Topologies for transformations 51
8．On range and null－space 52
9．The mean-ergodic theorem 54
Ⅲ HILBERT SPACE 57
1．Definition 57
2．Linear functionals 62
3．Orthonormal sets 65
4．Unbounded transformations and their adjoints 69
5．Projections 72
6．Resolutions of the identity 74
7．Unitary transformations 79
8．Examples and exercises 81
Ⅳ SPECTRAL THEORY OF LINEAR TRANSFORMATIONS 86
1．The setting 86
2．The spectrum 89
3．Integration procedures 91
4．The fundamental projections 92
5．A special case 99
6．The spectral radius 101
7．Analytic functions of operators 102
Ⅴ THE STRUCTURE OF SELF-ADJOINT TRANSFORMATIONS 106
1．Preliminary discussion 106
2．Positive operators 107
3．The point spectrum 110
4．The partition into pure types 112
5．The continuous spectrum 115
Ⅵ COMMUTATIVE BANACH ALGEBRAS 122
1．Introduction 122
2．Definitions and examples 123
3．The regular representation 125
4．Reducibility and idempotents 127
5．Algebras which are fields 128
6．Ideals 130
7．Quotient algebras 132
8．Homomorphisms and maximal ideals 136
9．The radical 141
10．The representation theory 143
11．Illustrative examples and applications 145
Selected References 153
Index 155