## Linear Operators, Part 1 |

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Page 40

For if I is maximal , then RI is a commutative ring with

ideals ; by what we showed earlier R ] I is a field . Conversely , if R / I is a field , it

contains no ideals and hence R has no ideals properly containing I . If R is a ring

...

For if I is maximal , then RI is a commutative ring with

**unit**which has no properideals ; by what we showed earlier R ] I is a field . Conversely , if R / I is a field , it

contains no ideals and hence R has no ideals properly containing I . If R is a ring

...

Page 41

Further , from the above we see that if M is a maximal ideal in a Boolean ring R

with

Boolean ring with a

be ...

Further , from the above we see that if M is a maximal ideal in a Boolean ring R

with

**unit**, then R / M is isomorphic with the field 02 . An important example of aBoolean ring with a

**unit**is the ring of subsets of a fixed set . More precisely , let Sbe ...

Page 485

Nelson Dunford, Jacob T. Schwartz. 8 THEOREM . ( Gantmacher ) An operator in

B ( X , Y ) is weakly compact if and only if its adjoint is weakly compact . PROOF .

Let T be weakly compact . Since the closed

Nelson Dunford, Jacob T. Schwartz. 8 THEOREM . ( Gantmacher ) An operator in

B ( X , Y ) is weakly compact if and only if its adjoint is weakly compact . PROOF .

Let T be weakly compact . Since the closed

**unit**sphere S * of Y * is Y - compact ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

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