## Linear Operators, Part 1 |

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Page 488

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

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 . Let 0 #ye Y and ...Page 489

since the range of U * is

restriction of y * to 3 , then x * = U ** . Hence , the range of U * is also

follows from the previous lemma that U_X = UX 3 . Hence , U has a

.

since the range of U * is

**closed**, æ * U * y * for some y * e Y * . If z * is therestriction of y * to 3 , then x * = U ** . Hence , the range of U * is also

**closed**. Itfollows from the previous lemma that U_X = UX 3 . Hence , U has a

**closed**range.

Page 513

( ii ) The range of U is

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 * . ( iv ) U * is one - to - one if and only if the range

of ...

( ii ) The range of U is

**closed**if there exists a constant K such that for any y in therange 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 * . ( iv ) U * is one - to - one if and only if the range

of ...

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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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