## Linear Operators, Part 1 |

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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

**function**fon S to X is said to be a u - null

**function**or , when u is understood , simply a null ...Page 105

Let f and g be

map of F ( S ) into F ( S ) . ... Proof . If In + f and gn → g , then , by Lemma 2 , \ In +

8n- ( 4 + g ) | Elin - 1 | + 8n - g → 0 and so f + g is a continuous

Let f and g be

**functions**on S to X. ( a ) The mapping f , g + f + g is a continuousmap of F ( S ) into F ( S ) . ... Proof . If In + f and gn → g , then , by Lemma 2 , \ In +

8n- ( 4 + g ) | Elin - 1 | + 8n - g → 0 and so f + g is a continuous

**function**of f and g .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero