## Linear Operators: General theory |

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Page 135

Let E be the union of an

Let E be the union of an

**arbitrary**sequence { En } of sets in S. Let E > 0 and for each n 1 , 2 , ... let the sequence { Em , n } have the properties Ε » , η « Σ , , En Cu Em , n Eu ( Em , n ) = û ( En ) + ɛ / 2n + 1 , m = 1 m = 1 00 ...Page 263

Since G is an

Since G is an

**arbitrary**open set containing F - G , we have Mi ( F ) S2 ( G ) + M ( F1-6 ) . If F is a closed set it follows from this inequality , by allowing G , to range over all open sets containing FF , that 41 ( F1 ) 5 ( FF1 ) + ...Page 476

N ( T ; A , E ) = { R R € B ( X , Y ) , ( T - R ) x ) < € , x € A } where A is an

N ( T ; A , E ) = { R R € B ( X , Y ) , ( T - R ) x ) < € , x € A } where A is an

**arbitrary**finite subset of X , and ε > 0 is**arbitrary**. Thus , in the strong topology , a generalized sequence { Ta } converges to T if and only if { T.x } ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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