## Linear Operators, Part 1 |

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Page 8

Hence , by Theorem 7 , there is a maximal well - ordered

= E for if x is in E but not in E , the ordering S ... 2 If a family & of

has the property that A € E if and only if every finite

...

Hence , by Theorem 7 , there is a maximal well - ordered

**subset**E , of E. Now E ,= E for if x is in E but not in E , the ordering S ... 2 If a family & of

**subsets**of a sethas the property that A € E if and only if every finite

**subset**of A belongs to E , then...

Page 335

Then L is complete and every

least upper bound of a countable

has an upper bound and let A , be the collection of least upper bounds of all ...

Then L is complete and every

**subset**A of L has a least upper bound which is theleast upper bound of a countable

**subset**of A. Proof . Let A be a**subset**of L whichhas an upper bound and let A , be the collection of least upper bounds of all ...

Page 440

totally ordered subfamily of A , the non - void set n A , is a closed extremal

of K which furnishes a lower bound for A 1. It follows by Zorn's lemma that A

contains a minimal element Ag . Suppose that A , contains two distinct points p

and q ...

totally ordered subfamily of A , the non - void set n A , is a closed extremal

**subset**of K which furnishes a lower bound for A 1. It follows by Zorn's lemma that A

contains a minimal element Ag . Suppose that A , contains two distinct points p

and q ...

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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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