## Linear Operators, Part 1 |

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Page 182

Thus d / du is

, E « . ... an additive set function My on E , the equation 4 ( 0 - ( E ) ) = P2 ( E )

additive ...

Thus d / du is

**defined**u - almost everywhere by the formula 2 ( E ) = S ( o ) ) ( ds ), E « . ... an additive set function My on E , the equation 4 ( 0 - ( E ) ) = P2 ( E )

**defines**an additive set function My on En . Moreover ( a ) if He is countablyadditive ...

Page 240

It is evident that if we

and uld ) = 0 , then a bounded function is E ... The space B ( S ) is

arbitrary set S and consists of all bounded scalar functions on S . The norm is

given ...

It is evident that if we

**define**the set function u on { by placing u ( E ) = 0 if E + $and uld ) = 0 , then a bounded function is E ... The space B ( S ) is

**defined**for anarbitrary set S and consists of all bounded scalar functions on S . The norm is

given ...

Page 534

function f

1 . anf ( x ) dx , n 20 , is a bounded map of L , into lą such that TT * is the map { an

} → { bn }

...

function f

**defined**on the interval ( 0 , 1 ) into the sequence { an }**defined**by an =1 . anf ( x ) dx , n 20 , is a bounded map of L , into lą such that TT * is the map { an

} → { bn }

**defined**by Show that T has norm Vr . Show that the map S of L2 ( 0 , 1 )...

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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