## Linear Operators: General theory |

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Page 42

e , then e = ax ' As an aid for the remainder of the proof , we demonstrate the following

e , then e = ax ' As an aid for the remainder of the proof , we demonstrate the following

**statement**: Let B , C B have the properties ( a ) if X , ye Bı , then xy e Bi ; ( b ) if x e Bı , then x = 0 ; then there is a homomorphism hy : B ...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 ) ( co ( A ) + co ( B ) .Page 447

**Statement**( b ) follows from the inequality a 2k ( x + = ( yı + y2 ) ) = k ( x + ayı ) + { ( x + ayz ) . 2**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality t ( x , y ) +1 ( x , y ) 21 ( , 0 ) = 0.t ( x , y ) = 0 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero