## Linear Operators: General theory |

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Page 487

Operators with Closed

Operators with Closed

**Range**It was observed in Lemma 2.8 that the closure of the**range**of an operator U e B ( X , Y ) consists of those vectors y such that y * UX = 0 implies y * y = 0. Or , in other words , UX = { y \ U * y * = 0 ...Page 488

It follows from the definition of U * that every element in its

It follows from the definition of U * that every element in its

**range**satisfies the stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a closed**range**, then UX = Y. PROOF .Page 489

since the

since the

**range**of U * is closed , ** U * y * for some y * e Y * . If z * is the restriction of y * to 3 , then x * U * z * . Hence , the**range**of U * is also closed . It follows from the previous lemma that U x = UX 3 .### 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