## Linear Operators, Part 1 |

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Page 464

( 3 ) For every | el , supwek Rf ( x ) < 00 ; moreover , each linear functional Ø on l '

which

= f ( x ) , jer for some X , € K . ( 4 ) If K , É < 0 , 0 a limit ordinal , is a monotone ...

( 3 ) For every | el , supwek Rf ( x ) < 00 ; moreover , each linear functional Ø on l '

which

**satisfies**inf Rf ( x ) = RO ( A ) S sup Rf ( x ) , ter XEK EK also**satisfies**0 ( 1 )= f ( x ) , jer for some X , € K . ( 4 ) If K , É < 0 , 0 a limit ordinal , is a monotone ...

Page 498

ΕεΣ ΕΕΣ Conversely , if w * ( . ) on to X *

operator T on X to L ( S , E , u ) whose norm

weakly compact if and only if x * ( • ) is countably additive on E in the strong

topology of X * .

ΕεΣ ΕΕΣ Conversely , if w * ( . ) on to X *

**satisfies**( i ) then ( ii ) defines anoperator T on X to L ( S , E , u ) whose norm

**satisfies**( iii ) . Furthermore T isweakly compact if and only if x * ( • ) is countably additive on E in the strong

topology of X * .

Page 557

Consequently , the product R , of all the factors ( a - hilai in R such that li € o ( T ) ,

still

hibi , where Bi = min ( di , v ( 2 ; ) ) ,

Consequently , the product R , of all the factors ( a - hilai in R such that li € o ( T ) ,

still

**satisfies**R ( T ) = 0 . In the same way , the product R , of all the factors ( a —hibi , where Bi = min ( di , v ( 2 ; ) ) ,

**satisfies**R ( T ) = 0 . Since any polynomial P ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero