## Linear Operators: General theory |

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Page 51

Q . E . D . 6 LEMMA . If a homomorphism of one topological group into another is

continuous anywhere , it is continuous . Proof . Let the homomorphism f : G H be

continuous at x , and let y eG . If V is a

...

Q . E . D . 6 LEMMA . If a homomorphism of one topological group into another is

continuous anywhere , it is continuous . Proof . Let the homomorphism f : G H be

continuous at x , and let y eG . If V is a

**neighborhood**of f ( y ) , then , by Lemma 2...

Page 56

Nelson Dunford, Jacob T. Schwartz. of the image of any

element 0 in X contains a

continuous function of a and b , there is a

MĒG .

Nelson Dunford, Jacob T. Schwartz. of the image of any

**neighborhood**G of theelement 0 in X contains a

**neighborhood**of the element 0 in Y . Since a - b is acontinuous function of a and b , there is a

**neighborhood**M of 0 such that M -MĒG .

Page 572

Conversely , let f ( T ) = 0 ; then , by Theorem 11 , 10 ( T ) ) = 0 . Let f be analytic

on a

that the sphere S ( Q , E ( Q ) ) CU . Since o ( T ) is compact , a finite set of

spheres ...

Conversely , let f ( T ) = 0 ; then , by Theorem 11 , 10 ( T ) ) = 0 . Let f be analytic

on a

**neighborhood**U of o ( T ) . For each a e O ( T ) , there is an e ( Q ) > 0 suchthat the sphere S ( Q , E ( Q ) ) CU . Since o ( T ) is compact , a finite set of

spheres ...

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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