## Linear Operators, Part 1 |

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Page 140

An

the forms : [ a , b ] = { sla SS sb } , [ a , b ) { sa ss < b } , ( a , b ] ... The number a is

called the left end point and b the right end point of any of these

An

**interval**is a set of points in the extended real number system which has one ofthe forms : [ a , b ] = { sla SS sb } , [ a , b ) { sa ss < b } , ( a , b ] ... The number a is

called the left end point and b the right end point of any of these

**intervals**.Page 141

i = 1 + 1 The argument may be repeated by defining & z = an , -c and choosing

appropriate points in the

every integer k 1 , 2 , ... , there are points a ;, bi with c < an sbn S ... Sa , sb , Sc + ...

i = 1 + 1 The argument may be repeated by defining & z = an , -c and choosing

appropriate points in the

**interval**( c , C + Eg ] . By induction , it is clear that forevery integer k 1 , 2 , ... , there are points a ;, bi with c < an sbn S ... Sa , sb , Sc + ...

Page 283

The proof will require the following two lemmas . 3 LEMMA . An almost periodic

function is bounded . PROOF OF LEMMA 3. Since an almost periodic function f is

continuous , the function f ( s ) has a maximum K on the

The proof will require the following two lemmas . 3 LEMMA . An almost periodic

function is bounded . PROOF OF LEMMA 3. Since an almost periodic function f is

continuous , the function f ( s ) has a maximum K on the

**interval**0 5 x SL ( 1 ) .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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 Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows 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 regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero