Introduction to the Analysis of Normed Linear SpacesThis is a basic course in functional analysis for senior undergraduate and beginning postgraduate students. The reader need only be familiarity with elementary real and complex analysis, linear algebra and have studied a course in the analysis of metric spaces; knowledge of integration theory or general topology is not required. The text concerns the structural properties of normed linear spaces in general, especially associated with dual spaces and continuous linear operators on normed linear spaces. The implications of the general theory are illustrated with a great variety of example spaces. |
Contents
NORMED LINEAR SPACE STRUCTURE AND EXAMPLES | 1 |
Exercises | 19 |
2 Classes of example spaces | 24 |
Exercises | 45 |
3 Orthonormal sets in inner product spaces | 50 |
Exercises | 64 |
SPACES OF CONTINUOUS LINEAR MAPPINGS | 67 |
Exercises | 86 |
13 Adjoint operators on Hilbert space | 181 |
Exercises | 193 |
14 Projection operators | 196 |
Exercises | 204 |
15 Compact operators | 206 |
Exercises | 214 |
Chapter VI SPECTRAL THEORY | 217 |
Exercises | 221 |
5 The shape of the dual | 92 |
Exercises | 110 |
THE EXISTENCE OF CONTINUOUS LINEAR FUNCTIONALS | 113 |
Exercises | 120 |
7 The natural embedding and reflexivity | 122 |
Exercises | 130 |
8 Subreflexivity | 132 |
Exercises | 138 |
Chapter IV THE FUNDAMENTAL MAPPING THEOREMS FOR BANACH SPACES | 139 |
Exercises | 150 |
10 The Open Mapping and Closed Graph Theorems | 153 |
Exercises | 158 |
11 The Uniform Boundedness Theorem | 163 |
Exercises | 169 |
Chapter V TYPES OF CONTINUOUS LINEAR MAPPINGS | 171 |
Exercises | 179 |
17 The spectrum of a continuous linear operator | 222 |
Exercises | 227 |
18 The spectrum of a compact operator | 230 |
Exercises | 235 |
19 The Spectral Theorem for compact normal operators on Hilbert space | 237 |
Exercises | 241 |
20 The Spectral Theorem for compact operators on Hilbert space | 243 |
Exercises | 250 |
APPENDIX | 252 |
A2 Numerical equivalence | 254 |
A3 Hamel basis | 256 |
Historical notes | 258 |
List of symbols | 269 |
List of spaces | 271 |
| 272 | |
Common terms and phrases
Baire space Banach space bounded Cauchy sequence characterisation closed graph closed linear subspace closed unit ball compact operator complex Hilbert space complex numbers conclude continuous functions continuous linear functional continuous linear mapping continuous linear operator convex Corollary countable deduce defined Definition dense dimensional normed linear e₁ eigenvalues element equivalent norms Euclidean Example exists f(ek f₁ f₂ finite dimensional normed functional f generalisation Given a continuous Given a normed Hahn-Banach Theorem Hamel basis Hilbert space hyperplane Il-ll Il·l implies inequality inner product space isometrically isomorphic ker f linear subspace M₁ metric space nonempty norm II-II normal operator normed linear space one-to-one orthogonal projection positive operator projection operator Proof proper closed linear properties Prove real numbers reflexive Remark scalar Schauder basis self-adjoint operators space H Spectral Theorem spectrum subset theory topological isomorphism unique x₁