## 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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The Interior

The Interior

**Mapping**Principle 3. The Hahn - Banach Theorem 4. Exercises 5. Notes and remarks 49 55 58 70 79 95 III . Integration and Set Functions 1.Page 1

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Three Basic Principles of Linear Analysis | 49 |

Integration and Set Functions | 95 |

The VitaliHahnSaks Theorem and Spaces of Measures | 155 |

Relativization of Set Functions | 164 |

Exercises | 168 |

Operators and Their Adjoints | 475 |

Adjoints | 478 |

Projections | 480 |

Weakly Compact Operators | 482 |

Compact Operators | 485 |

Operators with Closed Range | 487 |

Representation of Operators in CS | 489 |

The Representation of Operators in a Lebesgue Space | 498 |

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 4P | 281 |

SPECTRAL THEORY | 282 |

The Spaces LS E u | 285 |

Spaces of Set Functions | 305 |

Vector Valued Measures | 318 |

The Space TMS EM | 329 |

Functions of Bounded Variation | 337 |

Exercises | 338 |

Exercises on Orthogonal Series and Analytic Functions | 357 |

Tabulation of Results | 372 |

BAlgebras | 396 |

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 Vetrizability Lnbounded Sets 125 | 430 |

Exercises | 436 |

Extremal Points | 439 |

Tangent Functionals | 445 |

Fixed Point Theoremis | 453 |

Exercises | 457 |

Notes and Remarks | 460 |

Exercises | 511 |

The Riesz Convexity Theorem | 520 |

Exercises on Inequalities | 526 |

Notes and Remarks | 538 |

General Spectral Theory 355 | 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 |

Bounded Normal Operators in Hilbert Space | 611 |

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

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces Borel bounded called clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved reflexive regular respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique valued vector weak weakly compact zero