## Linear Operators: General theory |

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Page 36

A linear

group X together with an operation m : 0 X X → X , written as mía , x ) = ax , which

satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , ( ii ) ( a + b ) x = ax ...

A linear

**vector**space , linear space , or**vector**space over a field Ø is an additivegroup X together with an operation m : 0 X X → X , written as mía , x ) = ax , which

satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , ( ii ) ( a + b ) x = ax ...

Page 250

PROOF . If y * = 0 let y = 0. If y * # 0 the set M = { xy * x = 0 is a proper closed

linear manifold in H and its orthocomplement N contains , by Lemma 4 , a

yı 0. Let y y * ( y ) / \ y1 / . For an arbitrary

yı ) ...

PROOF . If y * = 0 let y = 0. If y * # 0 the set M = { xy * x = 0 is a proper closed

linear manifold in H and its orthocomplement N contains , by Lemma 4 , a

**vector**yı 0. Let y y * ( y ) / \ y1 / . For an arbitrary

**vector**æ in H the**vector**x- ( y * x ) / ( y *yı ) ...

Page 318

last section permit us to develop a more satisfactory theory of

countably additive set functions ( briefly ,

able ...

**Vector**Valued Measures The theorems on spaces of set functions proved in thelast section permit us to develop a more satisfactory theory of

**vector**valuedcountably additive set functions ( briefly ,

**vector**valued measures ) than we wereable ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element 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 meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero