## Linear Operators: General theory |

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Page 28

D -> X be a generalized sequence of elements in a metric space X. We call / a

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

that g(/(p), /(<?)) < £ if p 22 d0, g 22 d0. 5 Lemma. If f is a generalized Cauchy ...

D -> X be a generalized sequence of elements in a metric space X. We call / a

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

**exists**a d0e D, suchthat g(/(p), /(<?)) < £ if p 22 d0, g 22 d0. 5 Lemma. If f is a generalized Cauchy ...

Page 362

Under the hypotheses of Exercise 87, show that there

ffifixtynWdx if and only if the functions 2£—»^»iaw<Ma:)- m ^ 1, are uniformly

bounded and equicontinuous. 89 Let {an}, — oo <n< + oo, be a bounded

sequence of ...

Under the hypotheses of Exercise 87, show that there

**exists**/ in C with an =ffifixtynWdx if and only if the functions 2£—»^»iaw<Ma:)- m ^ 1, are uniformly

bounded and equicontinuous. 89 Let {an}, — oo <n< + oo, be a bounded

sequence of ...

Page 683

from Lemma 10 and Corollary 5.3 that the limit m — lim //„

Corollary 5.2, m(9P-1c) = m(e), so that the map T : /(•) -*□/(?>(")) as an operator

in the space L^{S, Z, m) has its norm \T\X = 1 (Lemma 5.7). Now let / be a

bounded ...

from Lemma 10 and Corollary 5.3 that the limit m — lim //„

**exists**in ca(Z, fi). ByCorollary 5.2, m(9P-1c) = m(e), so that the map T : /(•) -*□/(?>(")) as an operator

in the space L^{S, Z, m) has its norm \T\X = 1 (Lemma 5.7). Now let / be a

bounded ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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