## Linear Operators, Part 1 |

### From inside the book

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Page 2

The intersection , or product , of the sets a in A is the set of all x in U A which are

elements of every a e A . If A = { a , b , . . . , c } we will sometimes write the

U A as a Ubu . . . Uc , and the intersection n A as a non . . . nc , or simply as ab ...

The intersection , or product , of the sets a in A is the set of all x in U A which are

elements of every a e A . If A = { a , b , . . . , c } we will sometimes write the

**union**U A as a Ubu . . . Uc , and the intersection n A as a non . . . nc , or simply as ab ...

Page 10

B . TOPOLOGICAL PRELIMINARIES 4 . Definitions and Basic Properties 1

DEFINITION . A family t of subsets of a set X is called a topology in X if t contains

the void set ¢ , the set X , the

intersection ...

B . TOPOLOGICAL PRELIMINARIES 4 . Definitions and Basic Properties 1

DEFINITION . A family t of subsets of a set X is called a topology in X if t contains

the void set ¢ , the set X , the

**union**of every one of its subfamilies , and theintersection ...

Page 96

U An ) = 4 ( A2 ) + ( A2 ) + . . . tu ( An ) , for every finite family { A , , . . . , An } of

disjoint subsets of t whose

function , let S = ( 0 , 1 ) and let be the family of intervals I = ( a , b ) , 0 Sa < b < 1 ,

with ...

U An ) = 4 ( A2 ) + ( A2 ) + . . . tu ( An ) , for every finite family { A , , . . . , An } of

disjoint subsets of t whose

**union**is in t . For an example of a finitely additive setfunction , let S = ( 0 , 1 ) and let be the family of intervals I = ( a , b ) , 0 Sa < b < 1 ,

with ...

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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