## Linear Operators: General theory |

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Page 245

Every linear operator on a

Every linear operator on a

**finite**dimensional normed linear space is continuous . Proof . Let { b , ... , bn } be a Hamel basis for the**finite**dimensional normed linear space X so that every x in X has a unique representation in the ...Page 246

Then the dimension of X ** is

Then the dimension of X ** is

**finite**, and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is**finite**. Hence , from the first part of this proof , X and X * have the same dimension .Page 290

It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is o -

It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is o -

**finite**, and let En be an increasing sequence of measurable sets of**finite**measure whose union is S. Using the theorem for L ( En ) = L ( En ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero