## Linear Operators: General theory |

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Í : D → X be a generalized sequence of elements in a metric space X. We call f a generalized Cauchy sequence in X , if , for each x > 0 , there

Í : D → X be a generalized sequence of elements in a metric space X. We call f a generalized Cauchy sequence in X , if , for each x > 0 , there

**exists**a d , e D , such that g ( / ( p ) , f ( q ) ) < ε if p 2 do , q 2 do . 5 LEMMA .Page 353

Let M , be the space of all bounded 2 - measurable functions f defined on R such that lim f ( x )

Let M , be the space of all bounded 2 - measurable functions f defined on R such that lim f ( x )

**exists**. Putting lub \ | ( x ) ] , 2-0 OSE < 00 show that M , is a B - space . Let K ( x , y ) be a 2x2 measurable function on RX R.Page 362

Under the hypotheses of Exercise 37 , show that there

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with an 5347 ( x ) & n ( x ) dx if and only if the functions Ene - a m 2 1 , are uniformly bounded and equicontinuous . 39 Let { an ) , 0 < n < too , be a bounded ...### 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