## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 70

Page 495

We now prove that B , is an algebra under the natural product ( 1g ) ( s ) f ( s ) g (

s ) , s € S. To see this , let h be a fixed element of C ( S ) and let B ( h ) = { fe B. th

B , } . Evidently B ( h ) C Bo , and it is clear that B ( h )

We now prove that B , is an algebra under the natural product ( 1g ) ( s ) f ( s ) g (

s ) , s € S. To see this , let h be a fixed element of C ( S ) and let B ( h ) = { fe B. th

B , } . Evidently B ( h ) C Bo , and it is clear that B ( h )

**satisfies**properties ( i ) ...Page 498

on to X *

norm

countably additive on in the strong topology of X * . Proof . If , for E in E , the

functional x ...

on to X *

**satisfies**( i ) then ( ii ) defines an operator T on X to L ( S , E , ” ) whosenorm

**satisfies**( iii ) . Furthermore T is weakly compact if and only if x * ( ) iscountably additive on in the strong topology of X * . Proof . If , for E in E , the

functional x ...

Page 557

... zero polynomial R exists such that R ( T ) Let R be factored as R ( 2 ) = B119–1

( 2–2 ; ) . If his o ( T ) , then ( T - 2 ; 1 ) x = 0 implies x = 0. Consequently , the

product R , of all the factors ( 2–2 ; ) " in R such that li € ( T ) , still

0.

... zero polynomial R exists such that R ( T ) Let R be factored as R ( 2 ) = B119–1

( 2–2 ; ) . If his o ( T ) , then ( T - 2 ; 1 ) x = 0 implies x = 0. Consequently , the

product R , of all the factors ( 2–2 ; ) " in R such that li € ( T ) , still

**satisfies**R ( T ) =0.

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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