## Linear Operators, Part 1 |

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Page 68

The space X is said to be weakly complete if every weak Cauchy sequence has a

weak

such a way that they become Hausdorff spaces in which the notion of

convergence ...

The space X is said to be weakly complete if every weak Cauchy sequence has a

weak

**limit**. In Chapter V , a topology is introduced in certain linear spaces insuch a way that they become Hausdorff spaces in which the notion of

convergence ...

Page 126

We define the

En = Un Em , lim sup En = nu Em . in n = 1 men If lim inf , En = lim sup , En , { En }

is said to be convergent , and we write the common value of the

...

We define the

**limit**inferior and the**limit**superior of { En } by the equations lim infEn = Un Em , lim sup En = nu Em . in n = 1 men If lim inf , En = lim sup , En , { En }

is said to be convergent , and we write the common value of the

**limit**inferior and...

Page 293

It follows from Lemma II . 3 . 27 that K is bounded . If the integrals Sel ( s ) u ( ds )

are not countably additive uniformly with respect to f in K there is a number a > 0 ,

a decreasing sequence En e { with a void

It follows from Lemma II . 3 . 27 that K is bounded . If the integrals Sel ( s ) u ( ds )

are not countably additive uniformly with respect to f in K there is a number a > 0 ,

a decreasing sequence En e { with a void

**limit**, and functions in in K such that ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero