## Linear Operators: General theory |

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Page 289

... y * ( g ) when g and x * are connected , as in Theorem 1 , by the formula * * 1 =

st (

... y * ( g ) when g and x * are connected , as in Theorem 1 , by the formula * * 1 =

st (

**s**) g (**s**) u ( ds ) , - FEL , Applying Theorem 1 once more , this time to L * and**Lp**, we find there exists an he L , such that y * 1 = Ss h (**s**) f (**s**) u ( ds ) , felg .Page 524

Then log \ / \ 1 / a is a convex function of a , o sa s 1 . PROOF . If \ tl / a = too for all

a , 0 < a < 1 , the conclusion is trivial , so we suppose that f €

some po . By Lemma III . 8 . 5 , we may suppose that S is o - finite . Let Lo denote

...

Then log \ / \ 1 / a is a convex function of a , o sa s 1 . PROOF . If \ tl / a = too for all

a , 0 < a < 1 , the conclusion is trivial , so we suppose that f €

**Lp**(**S**, E , u ) forsome po . By Lemma III . 8 . 5 , we may suppose that S is o - finite . Let Lo denote

...

Page 527

... where 1 / p + 1 / 9 = 1 / r , and that \ hlo s Wilga thereby obtaining Hölder ' s

inequality as the special case 1 / p + 1 / 9 = 1 . ( p , q , r 2 1 . ) 2 Generalize

Exercise 1 to show that if h ( s ) = fı ( s ) . . . In ( s ) , tie

Pi = 1 / 1 ...

... where 1 / p + 1 / 9 = 1 / r , and that \ hlo s Wilga thereby obtaining Hölder ' s

inequality as the special case 1 / p + 1 / 9 = 1 . ( p , q , r 2 1 . ) 2 Generalize

Exercise 1 to show that if h ( s ) = fı ( s ) . . . In ( s ) , tie

**Lp**, (**S**, E , u ) , and n = 1 /Pi = 1 / 1 ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 |

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