## Linear Operators: General theory |

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Page 181

The v ( u ) -integrability of g ( :) ] , however , follows from Theorem 4 just as in the

argument used in the proof of Corollary 5 it is seen that it will be sufficient to

prove ...

The v ( u ) -integrability of g ( :) ] , however , follows from Theorem 4 just as in the

**preceding**argument . Thus only formula [ * ] remains to be verified . By theargument used in the proof of Corollary 5 it is seen that it will be sufficient to

prove ...

Page 396

Characterization of Ly and Lp . In the

the characterization of the space L , as a concrete representation of the abstract L

- spaces . Bohnenblust [ 1 ] has given a very interesting characterization of the L ...

Characterization of Ly and Lp . In the

**preceding**paragraph we have discussedthe characterization of the space L , as a concrete representation of the abstract L

- spaces . Bohnenblust [ 1 ] has given a very interesting characterization of the L ...

Page 684

Now conversely if ( 2 ) holds then the set function m in the proof of the

theorem satisfies the inequality m ( e ) skule ) and thus the u - integrable function

f is also m - integrable . It was observed in the

...

Now conversely if ( 2 ) holds then the set function m in the proof of the

**preceding**theorem satisfies the inequality m ( e ) skule ) and thus the u - integrable function

f is also m - integrable . It was observed in the

**preceding**proof that the limit g ( s )...

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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