## Linear Operators: General theory |

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Page 297

If Ei , i = 1 , 2 , ... , n are disjoint

If Ei , i = 1 , 2 , ... , n are disjoint

**sets**in & with 2-1 / ( E ; ) i Σ , ( ) > 21 - € , let ( 1 , ... , Chun be scalars with oil = 1 , 02 ( E ; ) = 12 ( E ) , and define f = 2-1 Xe ,, where Xe is the characteristic**function**of i ...Page 329

... shall be concerned here with a set S , a o - field of its subsets , and a scalar valued countably additive

... shall be concerned here with a set S , a o - field of its subsets , and a scalar valued countably additive

**set function**... E , u ) will be used for the set of all scalar valued functions on S which are totally u - measurable ( cf.Page 350

69 Show that the family of characteristic

69 Show that the family of characteristic

**functions**of**sets**in ? is fundamental in L ( S , E , u ) . 70 The**set**of all vectors x = { nn } with inn . 51 / n in real l , is called the Hilbert cube . Show that this**set**is compact in la .### 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