A Course in Functional AnalysisSpringer, 09.03.2019 - 400 Seiten Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts. |
Inhalt
| 1 | |
| 7 | |
4 Orthonormal Sets of Vectors and Bases | 14 |
CHAPTER II | 26 |
3 Projections and Idempotents Invariant and Reducing Subspaces | 36 |
5 The Diagonalization of Compact SelfAdjoint Operators | 46 |
7 The Spectral Theorem and Functional Calculus for Compact Normal | 54 |
8 Unitary Equivalence for Compact Normal Operators | 60 |
2 The BanachStone Theorem | 171 |
3 Compact Operators | 173 |
4 Invariant Subspaces | 178 |
5 Weakly Compact Operators | 183 |
CHAPTER VII | 187 |
2 Ideals and Quotients | 191 |
3 The Spectrum | 195 |
4 The Riesz Functional Calculus | 199 |
2 Linear Operators on Normed Spaces | 67 |
3 Finite Dimensional Normed Spaces 69 | 69 |
4 Quotients and Products of Normed Spaces | 70 |
5 Linear Functionals | 73 |
6 The HahnBanach Theorem | 77 |
Banach Limits | 82 |
Runges Theorem | 83 |
Ordered Vector Spaces | 86 |
10 The Dual of a Quotient Space and a Subspace | 88 |
11 Reflexive Spaces | 89 |
12 The Open Mapping and Closed Graph Theorems | 90 |
13 Complemented Subspaces of a Banach Space | 93 |
14 The Principle of Uniform Boundedness | 95 |
CHAPTER IV | 99 |
2 Metrizable and Normable Locally Convex Spaces | 105 |
3 Some Geometric Consequences of the HahnBanach Theorem | 108 |
4 Some Examples of the Dual Space of a Locally Convex Space | 114 |
5 Inductive Limits and the Space of Distributions | 116 |
CHAPTER V | 124 |
2 The Dual of a Subspace and a Quotient Space | 128 |
3 Alaoglus Theorem | 130 |
4 Reflexivity Revisited | 131 |
5 Separability and Metrizability | 134 |
The StoneČech Compactification | 137 |
7 The KreinMilman Theorem | 141 |
The StoneWeierstrass Theorem | 145 |
9 The Schauder Fixed Point Theorem | 149 |
10 The RyllNardzewski Fixed Point Theorem | 151 |
Haar Measure on a Compact Group | 154 |
12 The KreinSmulian Theorem | 159 |
13 Weak Compactness | 163 |
CHAPTER VI | 166 |
5 Dependence of the Spectrum on the Algebra | 205 |
6 The Spectrum of a Linear Operator | 208 |
7 The Spectral Theory of a Compact Operator | 214 |
8 Abelian Banach Algebras | 218 |
9 The Group Algebra of a Locally Compact Abelian Group | 223 |
CHAPTER VIII | 232 |
2 Abelian CAlgebras and the Functional Calculus in CAlgebras | 236 |
3 The Positive Elements in a CAlgebra | 240 |
4 Ideals and Quotients of CAlgebras | 245 |
5 Representations of CAlgebras and the GelfandNaimarkSegal Construction | 248 |
CHAPTER IX | 255 |
2 The Spectral Theorem | 262 |
3 StarCyclic Normal Operators | 268 |
4 Some Applications of the Spectral Theorem | 271 |
5 Topologies on BH | 274 |
6 Commuting Operators | 276 |
7 Abelian von Neumann Algebras | 281 |
The Conclusion of the Saga | 285 |
9 Invariant Subspaces for Normal Operators | 290 |
A Complete Set of Unitary Invariants | 293 |
CHAPTER X | 303 |
2 Symmetric and SelfAdjoint Operators | 308 |
3 The Cayley Transform | 316 |
4 Unbounded Normal Operators and the Spectral Theorem | 319 |
5 Stones Theorem | 327 |
6 The Fourier Transform and Differentiation | 334 |
7 Moments | 343 |
CHAPTER XI | 347 |
APPENDIX | 369 |
Bibliography | 384 |
Andere Ausgaben - Alle anzeigen
Häufige Begriffe und Wortgruppen
A₁ abelian assume ball Banach algebra Banach space Borel bounded linear bounded operator C*-algebra closed Co(X compact normal operator compact operator compact subset continuous function converges convex subset Corollary countable defined Definition denoted dense dim ker Example Exercise extreme point fact finite dimensional Fredholm functional calculus functions f h in H h₁ h₂ Hence Hilbert space homomorphism implies invariant subspace invertible isometrically isomorphic ker(A L²(µ Lemma linear functional linear manifold locally compact Math measure space metric normed space Note orthogonal orthonormal polynomial PROOF ran(A reader reflexive representation result self-adjoint operator semi-Fredholm semi-Fredholm operator seminorm sequence space and let spectral measure Spectral Theorem statements are equivalent Suppose surjective symmetric operator Theory topology unique unitary vector space Verify the statements weak-star weakly compact y₁