## Linear Operators, Part 1 |

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Page 186

... in proving uniqueness in Lemma 1 goes through without change in this case .

Q . E . D . 3 . DEFINITION . The

Theorem 2 is called the product

plin ) .

... in proving uniqueness in Lemma 1 goes through without change in this case .

Q . E . D . 3 . DEFINITION . The

**measure space**( S , E , u ) constructed inTheorem 2 is called the product

**measure space**of the**measure spaces**( Sn , En ,plin ) .

Page 188

Q . E . D . As in the case of finite

, Eis ui ) the product

Q . E . D . As in the case of finite

**measure spaces**we shall call the**measure****space**( S , & , u ) constructed in Corollary 6 from the o - finite**measure spaces**( Si, Eis ui ) the product

**measure space**and write ( S , E , u ) = P . – 1 ( Si , Eis Mil .Page 725

in - 1 lim sup - Eu ( 9 - ie ) < Kule ) n + N ; = 0 for each set e of finite u - measure . (

Hint . Consider the map 1 ( s ) + x ( s ) / ( ps ) for each A € £ with u ( A ) < 00 . ) 37

Let ( S , E , u ) be a positive

in - 1 lim sup - Eu ( 9 - ie ) < Kule ) n + N ; = 0 for each set e of finite u - measure . (

Hint . Consider the map 1 ( s ) + x ( s ) / ( ps ) for each A € £ with u ( A ) < 00 . ) 37

Let ( S , E , u ) be a positive

**measure space**, and T a non - negative linear ...### 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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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero