## Linear Operators: General theory |

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**Statement**( b ) is self - evident . Since A is closed , ( Lemma 4 ) , it follows that Ā = Ā . The set A UB contains A and is closed . Thus À UB2 Ā , and similarly A UB2 B. Therefore Ā UB 2 ĀU B. Conversely , Ā UB is closed ( Lemma 4 ) ...Page 415

... ( iv ) If CO ( A ) is compact , then co ( A + B ) CO ( A ) + co ( B ) . PROOF . The first part of

... ( iv ) If CO ( A ) is compact , then co ( A + B ) CO ( A ) + co ( B ) . PROOF . The first part of

**statement**( i ) follows in an elementary fashion from Lemma 1.4 . Lemma 1.4 also shows that co ( A + B ) C co ( A ) + co ( B ) .Page 447

**Statement**( b ) follows from the inequality a 21 ( x + 2 ( Yu + y2 ) ) = f ( x + ayı ) + k ( x + ayz ) .**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality T ( & , -y ) +7 ( x , y ) = f ( x , 0 ) = 0 ) · 7 ( 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