## Linear Operators: General theory |

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Page 245

An n - dimensional B - space is equivalent to En . 4 COROLLARY . Every linear

operator on a

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

An n - dimensional B - space is equivalent to En . 4 COROLLARY . 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 linear space X ...Page 246

Then the dimension of X * * is finite , and , since X is equivalent to a subspace of

X * * ( II . 3 . 19 ) , the ... 11 ) of infinite dimensional F - spaces with zero

dimensional conjugate spaces . ... A

reflerive .

Then the dimension of X * * is finite , and , since X is equivalent to a subspace of

X * * ( II . 3 . 19 ) , the ... 11 ) of infinite dimensional F - spaces with zero

dimensional conjugate spaces . ... A

**finite dimensional**normed linear space isreflerive .

Page 587

If one of the projections has a

and dim EX = dim E X . PROOF . Consider the map EE , E restricted to EX . Since

E is the identity in EX and since , by hypothesis , EE , E - E < 1 , it is seen from ...

If one of the projections has a

**finite dimensional**range then so does the otherand dim EX = dim E X . PROOF . Consider the map EE , E restricted to EX . Since

E is the identity in EX and since , by hypothesis , EE , E - E < 1 , it is seen from ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### 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

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