## 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 = a /

bzt ...

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 = a /

bzt ...

Page 290

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

of measurable sets of

En ) = L ( En , E ( En ) , u ) , we obtain a sequence { gn } of functions in L. such ...

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 ) , u ) , we obtain a sequence { gn } of functions in L. such ...

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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero