## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 88

Page 51

Q.E.D. 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

Lemma 2 ...

Q.E.D. 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 / ( y ) , then , byLemma 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

MCG .

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 -MCG .

Page 572

Thus , by Cauchy's integral formula , f ( T ) = 0 . Conversely , let f ( T ) = 0 ; then ,

by Theorem 11 , flo ( T ) ) = 0 . Let f be analytic on a

For each de O ( T ) , there is an e ( a ) > 0 such that the sphere S ( a , e ( a ) ) CU .

Thus , by Cauchy's integral formula , f ( T ) = 0 . Conversely , let f ( T ) = 0 ; then ,

by Theorem 11 , flo ( T ) ) = 0 . Let f be analytic on a

**neighborhood**U of o ( T ) .For each de O ( T ) , there is an e ( a ) > 0 such that the sphere S ( a , e ( a ) ) CU .

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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