## Linear Operators: General theory |

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Page 28

1 : D → X be a generalized sequence of elements in a metric space X. We call f a

generalized Cauchy sequence in X , if , for each a > 0 , there

such that ot ( p ) , f ( 9 ) ) < ε if p 2 do , I do . 5 LEMMA . If f is a generalized Cauchy

...

1 : D → X be a generalized sequence of elements in a metric space X. We call f a

generalized Cauchy sequence in X , if , for each a > 0 , there

**exists**a d , e D ,such that ot ( p ) , f ( 9 ) ) < ε if p 2 do , I do . 5 LEMMA . If f is a generalized Cauchy

...

Page 362

Under the hypotheses of Exercise 37 , show that there

) n ( x ) dx if and only if the functions ooimnanon ( Q ) , m 2 1 , are uniformly

bounded and equicontinuous . 39 Let { an ) , 0 < n < +00 , be a bounded

sequence of ...

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with an S * ( x) n ( x ) dx if and only if the functions ooimnanon ( Q ) , m 2 1 , are uniformly

bounded and equicontinuous . 39 Let { an ) , 0 < n < +00 , be a bounded

sequence of ...

Page 683

from Lemma 10 and Corollary 5.3 that the limit m = lim Mn

Corollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

from Lemma 10 and Corollary 5.3 that the limit m = lim Mn

**exists**in ca ( , u ) . ByCorollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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