## Linear Operators: General theory |

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Page 51

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

...

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 ( c ) , Vf ( y - 1x )...

Page 56

of the image of any

b , there is a

so x e nM ...

of the image of any

**neighborhood**G of the element 0 in X contains a**neighborhood**of the element 0 in y . Since a - b is a continuous function of a andb , there is a

**neighborhood**M of 0 such that M - MCG . For every x e X , xin 0 , andso x e nM ...

Page 572

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

a

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

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

a

**neighborhood**U of o ( T ) . For each a e o ( T ) , there is an e ( a ) > 0 such thatthe sphere S ( Q , € ( a ) ) CU . Since o ( T ) is compact , a finite set of spheres S ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero