## Linear Operators: General theory |

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Page 245

Every linear operator on a

PROOF . Let { b1 , . . . , bn } be a Hamel basis for the

linear space X so that every x in X has a unique representation in the form x = ...

Every linear operator on a

**finite**dimensional normed linear space is continuous .PROOF . Let { b1 , . . . , bn } be a Hamel basis for the

**finite**dimensional normedlinear space X so that every x in X has a unique representation in the form x = ...

Page 290

Now suppose that ( S , E , u ) is o -

of measurable sets of

En ) = L ( En , E ( En ) , ) , we obtain a sequence { 8n } of functions in La such that

...

Now suppose that ( S , E , u ) is o -

**finite**, and let En be an increasing sequenceof measurable sets of

**finite**measure whose union is S . Using the theorem for L (En ) = L ( En , E ( En ) , ) , we obtain a sequence { 8n } of functions in La such that

...

Page 849

3 ( 126 )

metric space , III . 7 . 1 ( 158 ) , III . 9 . 6 ( 169 ) positive , III . 4 . 3 ( 126 ) product ,

of

3 ( 126 )

**finite**, III . 4 . 3 ( 126 ) Lebesgue extension of I11 . 5 . 18 ( 143 ) as ametric space , III . 7 . 1 ( 158 ) , III . 9 . 6 ( 169 ) positive , III . 4 . 3 ( 126 ) product ,

of

**finite**number of**finite**measure spaces , III . 11 . 3 ( 186 ) of**finite**number of o -**finite**...### What people are saying - Write a review

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group 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