## Linear Operators: General theory |

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Page 244

Finite

prototype of all n -

observed first that En is a B - space . The Minkowski inequality ( III . 3 . 3 ) shows

En to ...

Finite

**Dimensional**Spaces The space E ” , as will be seen presently , is theprototype of all n -

**dimensional**normed linear spaces , and hence it should beobserved first that En is a B - space . The Minkowski inequality ( III . 3 . 3 ) shows

En to ...

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An n -

operator on a finite

b1 , . . . , bn } be a Hamel basis for the finite

An n -

**dimensional**B - space is equivalent to En . 4 COROLLARY . Every linearoperator on a finite

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

**dimensional**normed linear space X ...Page 246

Then the

X * * ( II . 3 . 19 ) , the

spaces with zero

Then the

**dimension**of X * * is finite , and , since X is equivalent to a subspace ofX * * ( II . 3 . 19 ) , the

**dimension**of X is finite . ... 11 ) of infinite**dimensional**F -spaces with zero

**dimensional**conjugate spaces . The following corollary was ...### 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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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