## Linear Operators: General theory |

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**COROLLARY**. For every x # 0 in a normed linear space X , there is an æ * € X * with 1c * / = 1 and x * x 121 . PROOF . Apply Lemma 12 with Y = 0. The v * required in the present**corollary**may then be defined as a times the x * whose ...Page 188

The best known example of Theorem 2 and its

The best known example of Theorem 2 and its

**Corollary**6 is obtained by taking ( Si , Ei , Mi ) to be the Borel - Lebesgue measure on the real line for i = 1 , ... , n . Then S = P1-1 S ; is n - dimensional Euclidean space , and j = My X ...Page 422

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**COROLLARY**. Lett be a linear functional on the linear space X , and let l ' be a total subspace of X * . Then the following statements are equivalent : ( i ) f is in l ' ; ( ii ) f is l ' - continuous ; ( iii ) H , = { x } ( x ) = 0 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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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