## 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 ) , V ] ( y - 1x )...

Page 56

of the image of any

b , there is a

and so X e ...

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 ,and so X e ...

Page 572

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

on a

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

S ...

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

on a

**neighborhood**U of o ( T ) . For each Q E O ( T ) , there is an ela ) > 0 suchthat the sphere S ( Q , Ela ) ) 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 | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero