## Linear Operators, Part 1 |

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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.

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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