## Linear Operators: General theory |

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Page 40

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

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

**unit**which has no proper ideals ; by what we showed earlier RI is a field . Conversely , if RI is a field , it contains no ideals and hence R has no ideals properly containing I.Page 41

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

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 og An important example of a Boolean ring with a**unit**is the ring of subsets of a fixed set .Page 485

Since the closed

Since the closed

**unit**sphere $ * of Y * is Y - compact ( V.4.2 ) , it follows from Lemma 7 and Lemma 1.5.7 that T * S * is compact in the *** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 isometric 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 u-measurable uniform uniformly unique unit valued vector weak weakly compact zero