## Linear Operators: General theory |

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Page 136

( Hahn

set function u on a field E has a countably additive non - negative

the o - field determined by E . If u is o - finite on Ethen this

( Hahn

**extension**) Every countably additive non - negative extended real valuedset function u on a field E has a countably additive non - negative

**extension**tothe o - field determined by E . If u is o - finite on Ethen this

**extension**is unique .Page 143

The o - field 2 * is known as the Lebesgue

E , and the measure space ( S , £ * , u ) is the Lebesgue

space ( S , E , u ) . Expressions such as u - simple function , totally u ...

The o - field 2 * is known as the Lebesgue

**extension**( relative to u ) of the o - fieldE , and the measure space ( S , £ * , u ) is the Lebesgue

**extension**of the measurespace ( S , E , u ) . Expressions such as u - simple function , totally u ...

Page 554

the

[ 6 ] showed that this operation is linear and isometric for each closed linear ...

**Extension**of Linear Transformation . Taylor [ 1 ] studied conditions under whichthe

**extension**of linear functionals will be a uniquely defined operation . Kakutani[ 6 ] showed that this operation is linear and isometric for each closed linear ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero