## Linear Operators: General theory |

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ES 14

ES 14

**COROLLARY**. For every x # 0 in a normed linear space X , there is an x * € X * with 122 * 1 = 1 and x * x lxl . PROOF . Apply Lemma 12 with Y = 0. The x * required in the present**corollary**may then be defined as x | times the æ ...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 = P_ Si is n - dimensional Euclidean space , and u = MX .Page 662

2

2

**COROLLARY**. When the strong limit E = lim , A ( n ) erists it is a projection of X upon the manifold { x \ Tx = x } of fixed points of T. The complementary projection has the closure of ( I – T ) X for its range . PROOF .### 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