## Linear Operators: General theory |

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Page 20

A sequence { an } is said to be convergent if an →a for some a . A sequence { an

} in a metric space is a

...

A sequence { an } is said to be convergent if an →a for some a . A sequence { an

} in a metric space is a

**Cauchy sequence**if limm , n olam , an ) = 0 . If every**Cauchy sequence**is convergent , a metric space is said to be complete . The next...

Page 68

which converges weakly to a point in X . Every sequence { Xn } such that { x * xx }

is a

which converges weakly to a point in X . Every sequence { Xn } such that { x * xx }

is a

**Cauchy sequence**of scalars for each x * € X * is called a weak**Cauchy****sequence**. The space X is said to be weakly complete if every weak Cauchy ...Page 122

Let 1 < p < 0o ; let { In } be a sequence of functions in L ( S , E , u , X ) and let f be

a function on S to X . Then f is in L , and lin ... Assuming ( i ) , ( ii ) , and ( iii ) let gn

= \ ( ) , g = \ | ( • ) | P . We will first show that { n } is a

Let 1 < p < 0o ; let { In } be a sequence of functions in L ( S , E , u , X ) and let f be

a function on S to X . Then f is in L , and lin ... Assuming ( i ) , ( ii ) , and ( iii ) let gn

= \ ( ) , g = \ | ( • ) | P . We will first show that { n } is a

**Cauchy sequence**in L ( S ) .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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

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