## Linear Operators, Part 1 |

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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 .

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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