## Linear Operators, Part 1 |

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Results 1-3 of 74

Page 40

For if I is maximal , then R / I 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 R / I 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 oz . An important example of aBoolean ring with a

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

Page 458

5 If the closed

finite number of extremal points , then X is not isometrically isomorphic to the

conjugate of any B - space . 6 Let S be a topological space , and let C ( S ) be the

B ...

5 If the closed

**unit**sphere of an infinite dimensional B - space X contains only afinite number of extremal points , then X is not isometrically isomorphic to the

conjugate of any B - space . 6 Let S be a topological space , and let C ( S ) be the

B ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero