## Linear Operators, Part 1 |

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Page 488

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

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

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

**range**satisfies thestated 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

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

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

.

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

.

( 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 ... ( iv ) U * is one - to - one if and only if the**range**of U is dense in Y. ( v ) If U maps onto y , then U * has a continuous inverse.

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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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