## Linear Operators: General theory |

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Page 95

CHAPTER III Integration and Set

contrast to the terms real

real and complex refer to the range of the

CHAPTER III Integration and Set

**Functions**1 . Finitely Additive Set**Functions**Incontrast to the terms real

**function**, complex**function**, etc . , where the adjectivesreal and complex refer to the range of the

**function**, the term set**function**is ...Page 103

space of all

rarely the case and so a slight detour will be made . 3 DEFINITION . The

on S to X is said to be a u - null

space of all

**functions**which map S into X ( see 1 . 6 . 1 ) . Unfortunately , this israrely the case and so a slight detour will be made . 3 DEFINITION . The

**functions**on S to X is said to be a u - null

**function**or , when u is understood , simply a null ...Page 165

Since a My - simple

- measurable

it is evident that f is also a u - integrable simple

...

Since a My - simple

**function**is clearly u - simple , it follows immediately that a My- measurable

**function**is u - measurable . If f is a My - integrable simple**function**,it is evident that f is also a u - integrable simple

**function**, and that set ( s ) py ( ds )...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 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 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