## Linear Operators: General theory |

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Page 387

A detailed proof was

A detailed proof was

**given**by Fréchet [ 5 ; III . p . 441 ] . The theorem for L , [ 0 , 1 ] , 1 < p < 0o , was demonstrated by F. Riesz [ 2 ; p . 475 ] . In the case of a finite measure space the theorem was established by Nikodým [ 9 ...Page 392

tion with respect to a vector valued measure which is presented here is the one

tion with respect to a vector valued measure which is presented here is the one

**given**in Bartle , Dunford and Schwartz [ 1 ] . A similar procedure has been employed by Bartle [ 3 ] to obtain a Lebesgue - type integration theory where ...Page 729

proofs , valid in uniformly convex spaces , were

proofs , valid in uniformly convex spaces , were

**given**by G. Birkhoff [ 7 ] and F. Riesz [ 16 , 18 ] . Another proof , based on the interesting fact that a fixed point for a contraction in Hilbert space is also a fixed point for its ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume 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 domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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