## Linear Operators: General theory |

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Page 34

The

satisfy the following conditions : ( i ) a ( be ) = ( ab ) c , a , b , c e G ; ( ii ) there is an

...

The

**element**ab is called the product of a and b . The product ab is required tosatisfy the following conditions : ( i ) a ( be ) = ( ab ) c , a , b , c e G ; ( ii ) there is an

**element**e in G , called the identity or the unit of G , such that ae = ea = a for every...

Page 40

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 with unit e , then an

right , left ) regular in R in case R contains a ( right , left ) inverse y for x , i . e . , we

...

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 with unit e , then an

**element**æ in R is called (right , left ) regular in R in case R contains a ( right , left ) inverse y for x , i . e . , we

...

Page 335

Let L be a o - complete lattice in which every set of

ordered under the partial ordering of L is ... between

that a Cb and that each

Let L be a o - complete lattice in which every set of

**elements**of L which is well -ordered under the partial ordering of L is ... between

**elements**a , b in W to meanthat a Cb and that each

**element**x which is in b but not a is an upper bound for a .### What people are saying - Write a review

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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