A Course in Functional AnalysisSpringer Science & Business Media, 17 באפר׳ 2013 - 406 עמודים 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. |
תוכן
| 1 | |
5 Isomorphic Hilbert Spaces and the Fourier Transform | 19 |
CHAPTER II | 26 |
3 FiniteDimensional Normed Spaces | 71 |
4 Quotients and Products of Normed Spaces | 73 |
5 Linear Functionals | 76 |
6 The HahnBanach Theorem | 80 |
Banach Limits | 84 |
4 Invariant Subspaces | 182 |
4 Compact Operators | 187 |
Banach Algebras and Spectral Theory for Operators on a Banach Space 1 Elementary Properties and Examples | 191 |
2 Ideals and Quotients | 195 |
3 The Spectrum | 199 |
4 The Riesz Functional Calculus | 203 |
5 Dependence of the Spectrum on the Algebra | 210 |
6 The Spectrum of a Linear Operator | 213 |
Runges Theorem | 86 |
Ordered Vector Spaces | 88 |
10 The Dual of a Quotient Space and a Subspace | 91 |
11 Reflexive Spaces | 92 |
12 The Open Mapping and Closed Graph Theorems | 93 |
13 Complemented Subspaces of a Banach Space | 97 |
14 The Principle of Uniform Boundedness | 98 |
CHAPTER IV | 102 |
2 Metrizable and Normable Locally Convex Spaces | 108 |
3 Some Geometric Consequences of the HahnBanach Theorem | 111 |
4 Some Examples of the Dual Space of a Locally Convex Space | 117 |
5 Inductive Limits and the Space of Distributions | 119 |
CHAPTER V | 127 |
2 The Dual of a Subspace and a Quotient Space | 131 |
3 Alaoglus Theorem | 134 |
4 Reflexivity Revisited | 135 |
5 Separability and Metrizability | 138 |
The StoneČech Compactification | 140 |
7 The KreinMilman Theorem | 145 |
The StoneWeierstrass Theorem | 149 |
9 The Schauder FixedPoint Theorem | 153 |
10 The RyllNardzewski FixedPoint Theorem | 155 |
Haar Measure on a Compact Group | 158 |
12 The KreinSmulian Theorem | 163 |
13 Weak Compactness | 167 |
CHAPTER VI | 170 |
2 The BanachStone Theorem | 175 |
3 Compact Operators | 177 |
7 The Spectral Theory of a Compact Operator | 219 |
8 Abelian Banach Algebras | 222 |
9 The Group Algebra of a Locally Compact Abelian Group | 228 |
CHAPTER VIII | 237 |
2 Abelian CAlgebras and the Functional Calculus in CAlgebras | 242 |
3 The Positive Elements in a CAlgebra | 245 |
4 Ideals and Quotients for CAlgebras | 250 |
5 Representations of CAlgebras and the GelfandNaimarkSegal Construction | 254 |
CHAPTER IX | 261 |
2 The Spectral Theorem | 268 |
3 StarCyclic Normal Operators | 275 |
4 Some Applications of the Spectral Theorem | 278 |
5 Topologies on BH | 281 |
6 Commuting Operators | 283 |
7 Abelian von Neumann Algebras | 288 |
The Conclusion of the Saga | 292 |
9 Invariant Subspaces for Normal Operators | 297 |
A Complete Set of Unitary Invariants | 299 |
CHAPTER X | 310 |
2 Symmetric and SelfAdjoint Operators | 316 |
3 The Cayley Transform | 323 |
4 Unbounded Normal Operators and the Spectral Theorem | 326 |
CHAPTER XI | 353 |
APPENDIX | 375 |
Bibliography | 390 |
| 392 | |
מהדורות אחרות - הצג הכל
מונחים וביטויים נפוצים
A₁ abelian assume ball Banach algebra Banach space bijection Borel bounded linear C*-algebra closure Co(X compact operator compact space compact subset continuous function continuous seminorm converges convex subset Corollary countable defined Definition denoted dense dim ker dim ran Example Exercise extreme point f₁ fact finite functional calculus functions f h in H h₁ h₂ Hence Hilbert space homeomorphism implies invertible isometrically isomorphic isometry K₁ ker(A L²(µ Lemma Let f linear functional linear manifold linear span locally compact measure space metric N₁ normed space Note o-finite open subset orthonormal polynomial PROOF ran(A reader reflexive result self-adjoint operator seminorm separable sequence space and let spectral measure Spectral Theorem Suppose surjective unique unitary vector space Verify the statements weak topology weak-star weakly compact y₁