## Linear Operators: General theory |

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Page 162

We note that if X is the space of scalars ( real or complex ) it will usually be

omitted from notations like ca ( S ,

€ for an appropriate choice of the indices ( j , . . . , Olm . If

then ...

We note that if X is the space of scalars ( real or complex ) it will usually be

omitted from notations like ca ( S ,

**E**, X ) . Thus ca ( S ... + Mo ( Am ) Su ( EU F ) +€ for an appropriate choice of the indices ( j , . . . , Olm . If

**E**; = AE and**Fi**= A ; Fthen ...

Page 170

Show that this is no longer the case if TM is replaced by M . 14 Let & , be a

subfield of

€ > 0 there exist sets F , and F2 € Į , such that

< € .

Show that this is no longer the case if TM is replaced by M . 14 Let & , be a

subfield of

**E**, and My the restriction of u to £ . ... Call a set**E**€ { regular if for each€ > 0 there exist sets F , and F2 € Į , such that

**Fi**CE , F , CS -**E**, v ( u , S - F , - F2 )< € .

Page 334

If f , and fz are real or complex valued functions in M ( S ,

we write

the sense of Definition I . 2 . 1 . We note that the space TM ( S ,

If f , and fz are real or complex valued functions in M ( S ,

**E**, u ) and if f1 - 12 20 ,we write

**fi**2 ta or t2**fi**. This makes M ( S ,**E**, u ) into a partially ordered system inthe sense of Definition I . 2 . 1 . We note that the space TM ( S ,

**E**, u ) is also ...### 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