## Linear Operators: General theory |

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Page 162

Thus ca ( S , E ) is the space of all real or complex valued countably additive set functions defined on Σ . We have just

Thus ca ( S , E ) is the space of all real or complex valued countably additive set functions defined on Σ . We have just

**seen**that the spaces ba ( S , E ) and ca ( S , E ) are Bspaces . For some purposes it is useful to know that they ...Page 254

Thus by forming the chain Up , Ugg ... , Ug's Up . it is

Thus by forming the chain Up , Ugg ... , Ug's Up . it is

**seen**that vp is equivalent to vg . and thus that VB is in V. Since { vp } is a basis , the vector Wq has an expansion of the form un = Eb ( ug , VB ) Up , so that wq is in the ...Page 567

We have

We have

**seen**in the proof of Lemma 2 that if sul < | R ( 2 ; T ) | -1 , then a + u e g ( T ) . Hence d ( 2 ) 2 | R ( 2 ; T ) | -1 , from which the statements follow . Q.E.D. 4 LEMMA . The closed set o ( T ) is bounded and non - void .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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 isometric 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 u-measurable uniform uniformly unique unit valued vector weak weakly compact zero