## Linear Operators: General theory |

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Page 91

However , Kolmogoroff [ 1 ] proved that a topological linear space is

convex neighborhood of the origin . Wehausen [ 1 ] showed that if a topological

linear space has ...

However , Kolmogoroff [ 1 ] proved that a topological linear space is

**homeomorphic**to a normed linear space if and only if there exists a boundedconvex neighborhood of the origin . Wehausen [ 1 ] showed that if a topological

linear space has ...

Page 442

If Q is a compact Hausdorff space , and C ( Q ) is the B - space of all real or

complex continuous functions on Q , then the mapping a : 9 + * * of Q into a

subset Q of the extremal points of S * is a

C ( Q ) ...

If Q is a compact Hausdorff space , and C ( Q ) is the B - space of all real or

complex continuous functions on Q , then the mapping a : 9 + * * of Q into a

subset Q of the extremal points of S * is a

**homeomorphism**where C * ( Q ) has itsC ( Q ) ...

Page 443

Thus t is a

Q and u : R → R are

to Q . Thus , for any x e C ( Q ) and r e R , we have ( Tx ) ( r ) = y * ( Tx ) = ( T * y ...

Thus t is a

**homeomorphism**between R and Q . But since , by Lemma 7 , 2 : Q →Q and u : R → R are

**homeomorphisms**, T = 2 - 1 tu is a**homeomorphism**from Rto Q . Thus , for any x e C ( Q ) and r e R , we have ( Tx ) ( r ) = y * ( Tx ) = ( T * y ...

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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