## Linear Operators: General theory |

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Page 140

An

the forms : [ a , b ] = { sa Sssb } , [ a , b ) = { ssa s < b } , ( a , b ] { ssa ... 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 ] = { sa Sssb } , [ a , b ) = { ssa s < b } , ( a , b ] { ssa ... The number

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

**intervals**.Page 141

3 i = ni + 1 The argument may be repeated by defining Ez = choosing appropriate

points in the

1 , 2 , ... , there are points a ;, bi with c < an Sbne S ... Sansby Sc + & such that ...

3 i = ni + 1 The argument may be repeated by defining Ez = choosing appropriate

points in the

**interval**( c , c + Eg ] . By induction , it is clear that for every integer k =1 , 2 , ... , there are points a ;, bi with c < an Sbne S ... Sansby Sc + & such that ...

Page 283

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

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

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

SL ...

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

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

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

**interval**o 5 xSL ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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