## Linear Operators: General theory |

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Page 464

( 2 ) If xs , & < 0 , 0 a limit ordinal , is a transfinite sequence in K , then there is an

x , eK with lim inf Rf ( x3 ) $ t ( 20 ) Slim sup Rf ( xg ) , for fel . ( 3 ) For every f e l ' ,

sup . zek Rf ( x ) < 00 ; moreover , each linear functional 0 on l ' which

...

( 2 ) If xs , & < 0 , 0 a limit ordinal , is a transfinite sequence in K , then there is an

x , eK with lim inf Rf ( x3 ) $ t ( 20 ) Slim sup Rf ( xg ) , for fel . ( 3 ) For every f e l ' ,

sup . zek Rf ( x ) < 00 ; moreover , each linear functional 0 on l ' which

**satisfies**inf...

Page 498

E€ 2 Conversely , if x * ( ) on to X *

X to L ( S , E , u ) whose norm

and only if x * ( - ) is countably additive on { in the strong topology of X * . PROOF

...

E€ 2 Conversely , if x * ( ) on to X *

**satisfies**( i ) then ( ii ) defines an operator T onX to L ( S , E , u ) whose norm

**satisfies**( iii ) . Furthermore T is weakly compact ifand only if x * ( - ) is countably additive on { in the strong topology of X * . PROOF

...

Page 557

Consequently , the product R , of all the factors ( 9 – 2 ; ) xi in R such that hi e o (

T ) , still

- 2 ; Bi , where Pi = min ( lig v ( ? ; ) ) ,

...

Consequently , the product R , of all the factors ( 9 – 2 ; ) xi in R such that hi e o (

T ) , still

**satisfies**R ( T ) = 0 . In the same way , the product R , of all the factors ( 2- 2 ; Bi , where Pi = min ( lig v ( ? ; ) ) ,

**satisfies**R , ( T ) = 0 . Since any polynomial...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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