## Linear Operators: General theory |

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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 658

Thus , from the physical point of view , the time T involved in the experiment is

large enough to give a good value for the

which , in physical theories , is assumed to exist . Thus , the mathematician is led

to ...

Thus , from the physical point of view , the time T involved in the experiment is

large enough to give a good value for the

**limit**( iii ) lim = f ( d ( ) ) dt , TTM T Jowhich , in physical theories , is assumed to exist . Thus , the mathematician is led

to ...

Page 683

3 that the

e ) , so that the map T : f ( - ) → ( 4 ( : ) ) as an operator in the space L ( S , E , m )

has its norm Ti = 1 ( Lemma 5 . 7 ) . Now let f be a bounded u - measurable ...

3 that the

**limit**m = lim un exists in ca ( E , u ) . By Corollary 5 . 2 , m ( 9 - le ) = m (e ) , so that the map T : f ( - ) → ( 4 ( : ) ) as an operator in the space L ( S , E , m )

has its norm Ti = 1 ( Lemma 5 . 7 ) . Now let f be a bounded u - measurable ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero