## Linear Operators: General theory |

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A linear

A linear

**vector**space , linear space , or**vector**space over a field o is an additive group X together with an operation m : 0 X X → X , written as m ( a , x ) = ax , which satisfy the following four conditions : x i ( i ) a ( x + y ) ...Page 250

If y * # 0 the set M = { wy * x = 0 } x \ is a proper closed linear manifold in H and its orthocomplement » contains , by Lemma 4 , a

If y * # 0 the set M = { wy * x = 0 } x \ is a proper closed linear manifold in H and its orthocomplement » contains , by Lemma 4 , a

**vector**Yi # 0. Let y = ayı where ā y * ( y ) / | 41 | 4 . For an arbitrary**vector**æ in H the**vector**x- ...Page 318

**Vector**Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of**vector**valued countably additive set functions ( briefly ,**vector**valued measures ) than we ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad 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 semi-group 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