## Linear Operators: General theory |

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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 , æ * U * y * for some y * e Y * . If z * is the restriction of y * to 3 , then x * = U ** . Hence , the**range**of U * is also closed . It follows from the previous lemma that U_X = UX 3 .Page 513

( ii ) The

( ii ) The

**range**of U is closed if there exists a constant K such that for any y in the**range**there exists a solution of y Tæ such that x Skyl . ( iii ) U is one - to - one if the**range**of U * is dense in X * .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero