## Linear Operators: General theory |

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Page 3

We presuppose a familiarity with the real and complex number systems . By the

extended

and - 00 adjoined ; by the extended complex number system we mean the

complex ...

We presuppose a familiarity with the real and complex number systems . By the

extended

**real number**system we mean the**real numbers**with the symbols tooand - 00 adjoined ; by the extended complex number system we mean the

complex ...

Page 4

n - > If $ is the void subset of the

sup , + = inf $ . If A is an infinite set of

denotes the infimum of all numbers b with the property that only a finite set of ...

n - > If $ is the void subset of the

**real numbers**, it is conventional to take - 00 =sup , + = inf $ . If A is an infinite set of

**real numbers**, then the symbol lim sup Adenotes the infimum of all numbers b with the property that only a finite set of ...

Page 11

The topology of the extended

b ) , ( 6 , + 00 ] , where a and b are

extended complex numbers are treated similarly . 7 LEMMA . If B is a family of ...

The topology of the extended

**real numbers**has as a base the sets ( - 00 , a ) , ( a ,b ) , ( 6 , + 00 ] , where a and b are

**real numbers**. The complex numbers and theextended complex numbers are treated similarly . 7 LEMMA . If B is a family of ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter 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 Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero