## 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

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

Every linear operator on a

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

**finite**dimensional normed linearspace X so that every x in X has a unique representation in the form x = Qzba + ...

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

( See Decomposition ) definition , III.4.3 ( 126 )

extension of , III.5.18 ( 143 ) as a metric space , II1.7.1 ( 158 ) , III.9.6 ( 169 )

positive , III.4.3 ( 126 ) product , of

( 186 ) of ...

( See Decomposition ) definition , III.4.3 ( 126 )

**finite**, III.4.3 ( 126 ) Lebesgueextension of , III.5.18 ( 143 ) as a metric space , II1.7.1 ( 158 ) , III.9.6 ( 169 )

positive , III.4.3 ( 126 ) product , of

**finite**number of**finite**measure spaces , III.11.3( 186 ) of ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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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