## 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 e G. 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 e G. If V is a

**neighborhood**of f ( y ) , then , by Lemma 2 ( c ) , Vily - 1x ...Page 56

of the image of any

b , there is a

and so xe ...

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 , x / n +0 ,and so xe ...

Page 572

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

a

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

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

a

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero