## Linear Operators: General theory |

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Results 1-3 of 83

Page 7

... to some subset E , E E , and x Soy in the ordering o of that E . It is clear that if (

UE , S ' ) belongs to E it is an upper bound for En . It will now be

well - ordered and hence belongs to E . The statement « S ' x for xe UE is clear .

... to some subset E , E E , and x Soy in the ordering o of that E . It is clear that if (

UE , S ' ) belongs to E it is an upper bound for En . It will now be

**shown**that it iswell - ordered and hence belongs to E . The statement « S ' x for xe UE is clear .

Page 335

It will first be

we let W , be a totally ordered subset of W ( I . 22 ) and let c CUW . . Then , for

some a € W , cna is not void . Let x be the smallest element of cna and let y be

any ...

It will first be

**shown**that W satisfies the hypothesis of Zorn ' s lemma . To do thiswe let W , be a totally ordered subset of W ( I . 22 ) and let c CUW . . Then , for

some a € W , cna is not void . Let x be the smallest element of cna and let y be

any ...

Page 553

841 ] has

seen in Theorems 7 . 4 and 8 . 12 that in the spaces C and Li , FCCC W CP .

Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and P coincide . This is

not ...

841 ] has

**shown**that in Hilbert space C is a maximal two - sided ideal . We haveseen in Theorems 7 . 4 and 8 . 12 that in the spaces C and Li , FCCC W CP .

Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and P coincide . This is

not ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero