## Linear Operators: General theory |

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Page 18

A continuous one - to - one map from a

homeomorphism . PROOF . Let X be a

and f a one - to - one continuous function on X , with f ( X ) = Y . According to the ...

A continuous one - to - one map from a

**compact**space to a**Hausdorff space**is ahomeomorphism . PROOF . Let X be a

**compact**space , Y a**Hausdorff space**,and f a one - to - one continuous function on X , with f ( X ) = Y . According to the ...

Page 276

Suppose , in addition to the hypotheses of Theorem 18 , that the functions of A

distinguish between the points of S . Then there exists a

each ...

Suppose , in addition to the hypotheses of Theorem 18 , that the functions of A

distinguish between the points of S . Then there exists a

**compact Hausdorff****space**S , and a one - to - one embedding of S as a dense subset of S , such thateach ...

Page 516

39 Let S be a

Show that there is a regular countably additive non - negative measure u defined

for all Borel sets in S with the properties that u is not identically zero and u is ...

39 Let S be a

**compact Hausdorff space**and $ a continuous function on S to S .Show that there is a regular countably additive non - negative measure u defined

for all Borel sets in S with the properties that u is not identically zero and u is ...

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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