## Linear Operators: General theory |

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Page 48

The first occurrence of a maximum principle

principle ( as in Theorem 2 . 6 ) is in Hausdorff [ 1 ; p . 140 ] . Zorn [ 1 ] gave a

theorem essentially

...

The first occurrence of a maximum principle

**equivalent**to the well - orderingprinciple ( as in Theorem 2 . 6 ) is in Hausdorff [ 1 ; p . 140 ] . Zorn [ 1 ] gave a

theorem essentially

**equivalent**to Theorem 2 . 7 . A similar theorem is due to R . L...

Page 91

admits an

invariant metric and is complete under each invariant metric . Thus every

complete linear metric space can be metrized to be an F - space . Further , a

normed linear ...

admits an

**equivalent**metric under which it is complete , then G admits aninvariant metric and is complete under each invariant metric . Thus every

complete linear metric space can be metrized to be an F - space . Further , a

normed linear ...

Page 347

( b ) Show that L ( S , E , u ) is

collection of atoms of finite measure { En } in such that every measurable subset

of S - Um - 1 En is either an atom of infinite measure or a null set . 50 Show that ...

( b ) Show that L ( S , E , u ) is

**equivalent**to l if and only if there exists a countablecollection of atoms of finite measure { En } in such that every measurable subset

of S - Um - 1 En is either an atom of infinite measure or a null set . 50 Show that ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero