## Linear Operators: General theory |

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Page 170

Show that

Sv * ( u , E ) nor

there exist sets F , and F2 € Į , such that

Show that

**v**( M , E ) =**v**( u , E ) holds for every E € £ , but that neither**v*** ( M , E )Sv * ( u , E ) nor

**v*** ( u , E ) S**v*** ( un , E ) ... Call a set E € { regular if for each € > 0there exist sets F , and F2 € Į , such that

**Fi**CE , F , CS - E ,**v**( u , S - F , - F2 ) < € .Page 262

The function f is therefore the uniform limit of u - simple functions and , since

, S ) < 00 , f is u - integrable . ... 2 , there exists a set { f1 , . . . , In } of continuous

functions with o sti ( s ) 1 , and such that

The function f is therefore the uniform limit of u - simple functions and , since

**v**( u, S ) < 00 , f is u - integrable . ... 2 , there exists a set { f1 , . . . , In } of continuous

functions with o sti ( s ) 1 , and such that

**fi**( s ) = 0 if s ¢G ; and**fi**( s ) = 1 if s e C ; .Page 273

... define to = isant Voo . Vip , to Thus f ( u ) > F ( u ) - € for u e S . Since ts . . ( t ) =

F ( t ) , we have

u ) < F ( u ) + € , u e

...

... define to = isant Voo . Vip , to Thus f ( u ) > F ( u ) - € for u e S . Since ts . . ( t ) =

F ( t ) , we have

**fi**( t ) = F ( t ) , and hence there is a neighborhood**V**, of t with fo (u ) < F ( u ) + € , u e

**V**. Let**V**, . . . ,**V**. , cover S , and define f = f , ^ . . . ntti Since f :...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero