## Linear Operators: General theory |

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Page 488

If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a

If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a

**closed**range , then UX = Y. PROOF . Let 0 #ye Y and define I = { y * y * € Y * , y * y = 0 } . Then I ' is Y -**closed**in Y * . Suppose , for the moment , that U ...Page 489

since the range of U * is

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 range of U is

( 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 2 SK y . ( 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 | |

87 other sections not shown

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