## Linear Operators: General theory |

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Page 18

A continuous one - to - one map from a compact space to a

homeomorphism . PROOF . Let X be a compact space , Y 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 274

Let S be a compact

continuous functions on S . Let A be a closed subalgebra of C ( S ) which

contains the unit e and contains , with f , its complex conjugate f defined by F ( s )

= f ( $ ) .

Let S be a compact

**Hausdorff**space and C ( S ) be the algebra of all complexcontinuous functions on S . Let A be a closed subalgebra of C ( S ) which

contains the unit e and contains , with f , its complex conjugate f defined by F ( s )

= f ( $ ) .

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 compact

space S , and a one - to - one embedding of S as a dense subset of S , such that

each f ...

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 that

each f ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

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