Linear Operators, Part 1: General TheoryThis classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician—treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes. |
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Contents
Preliminary Concepts | 1 |
8 | 28 |
Algebraic Preliminaries | 34 |
Exercises | 46 |
Integration and Set Functions | 95 |
The Lebesgue Spaces | 119 |
Countably Additive Set Functions | 129 |
Integration with Respect to a Countably Additive Measure | 144 |
Exercises | 457 |
Notes and Remarks | 460 |
Operators and Their Adjoints | 475 |
Adjoints | 478 |
Projections | 480 |
Weakly Compact Operators | 482 |
Compact Operators | 485 |
Operators with Closed Range | 487 |
The VitaliHahnSaks Theorem and Spaces of Measures | 155 |
Relativization of Set Functions | 164 |
Exercises | 168 |
The RadonNikodým Theorem | 174 |
Product Measures | 183 |
Differentiation | 210 |
Exercises | 222 |
Functions of a Complex Variable | 224 |
Notes and Remarks | 232 |
Special Spaces | 237 |
A List of Special Spaces | 238 |
Finite Dimensional Spaces | 244 |
Hilbert Space | 247 |
The Spaces BS E and BS | 257 |
The Space CS | 261 |
The Space AP | 281 |
The Spaces LS Д µ | 285 |
Spaces of Set Functions | 305 |
Vector Valued Measures | 318 |
The Space TMS Σµ | 329 |
Functions of Bounded Variation | 337 |
Exercises | 338 |
Exercises on Orthogonal Series and Analytic Functions | 357 |
Tabulation of Results | 372 |
Convex Sets and Weak Topologies | 409 |
Linear Topological Spaces | 413 |
Weak Topologies Definitions and Fundamental Properties | 418 |
Weak Topologies Compactness and Reflexivity | 423 |
Weak Topologies Metrizability Unbounded Sets | 425 |
Weak Topologies Weak Compactness | 430 |
Exercises | 436 |
Extremal Points | 439 |
Tangent Functionals | 445 |
Fixed Point Theorems | 453 |
Representation of Operators in CS | 489 |
The Representation of Operators in a Lebesgue Space | 498 |
Exercises | 511 |
The Riesz Convexity Theorem | 520 |
Exercises on Inequalities | 526 |
Notes and Remarks | 538 |
General Spectral Theory | 555 |
Spectral Theory in a Finite Dimensional Space | 556 |
Exercises | 561 |
Functions of an Operator | 566 |
Spectral Theory of Compact Operators | 577 |
Exercises | 580 |
Perturbation Theory | 584 |
Tauberian Theory | 593 |
Exercises | 597 |
An Operational Calculus for Unbounded Closed Operators | 599 |
Exercises | 604 |
Notes and Remarks | 606 |
Applications | 613 |
Functions of an Infinitesimal Generator | 641 |
Exercises | 653 |
Ergodic Theory | 657 |
Mean Ergodic Theorems | 660 |
Pointwise Ergodic Theorems | 668 |
The Ergodic Theory of Continuous Flows | 684 |
Uniform Ergodic Theory | 708 |
Exercises on Ergodic Theory | 717 |
Notes and Remarks | 726 |
REFERENCES | 731 |
Sufficient Conditions XVII Algebras of Spectral Operators XVIII Unbounded Spectral Operators XIX Perturbations of Spectral Operators with Discre... | 735 |
NOTATION INDEX | 827 |
829 | |
837 | |
Common terms and phrases
A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote differential Doklady Akad E₁ element equation exists extension f₁ fn(s function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f lim sup linear functional linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space u-integrable u-measurable uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ