## Linear Operators: General theory |

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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

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

measurable subset of S - 1 , En is either an atom of infinite measure or a null set .

50 Show that no ...

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

**equivalent**to l , if and only if there exists acountable collection of atoms of finite measure { En } in such that every

measurable subset of S - 1 , En is either an atom of infinite measure or a null set .

50 Show that no ...

Page 663

... maps L ( S , E , u ) into L ( S , E , u ) and has the corresponding sequence { A (

n ) } bounded . Since the points in L ( S , E , u ) are not functions but classes of

LP ...

... maps L ( S , E , u ) into L ( S , E , u ) and has the corresponding sequence { A (

n ) } bounded . Since the points in L ( S , E , u ) are not functions but classes of

**equivalent**functions , it is seen that T may not be regarded as being defined onLP ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero