## Linear Operators: General theory |

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Page 387

A detailed proof was

, 1 < p < 0 , was demonstrated by F. Riesz [ 2 ; p . 475 ] . In the case of a finite

measure space the theorem was established by Nikodým [ 9 ; p . 132 ] and later

by ...

A detailed proof was

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

measure space the theorem was established by Nikodým [ 9 ; p . 132 ] and later

by ...

Page 392

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

employed by Bartle [ 3 ] to obtain a Lebesgue - type integration theory where both

...

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 beenemployed by Bartle [ 3 ] to obtain a Lebesgue - type integration theory where both

...

Page 729

proofs , valid in uniformly convex spaces , were

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 adjoint , was

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 adjoint , was

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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