## Linear Operators, Part 1 |

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Page 365

Show that the map : f + $ defined in

manner onto the closed subspace of L consisting of those F all of whose negative

Fourier coefficients vanish . 55 Using the notations of

...

Show that the map : f + $ defined in

**Exercise**53 maps H , in a linear one - onemanner onto the closed subspace of L consisting of those F all of whose negative

Fourier coefficients vanish . 55 Using the notations of

**Exercises**53 and 54 , show...

Page 371

86 Let p , f be as in

and let | be a function in Hg . Then there exists a function g in Ho such that g ( e )

= 1 for almost all 0 , LEH , , - = \ , and such that f / g has no zeros .

86 Let p , f be as in

**Exercise**85 . Then [ ( elo ) + 0 for almost all 0 . 87 Let p > 1and let | be a function in Hg . Then there exists a function g in Ho such that g ( e )

= 1 for almost all 0 , LEH , , - = \ , and such that f / g has no zeros .

Page 531

This is the vector form of

next set of inequalities are all variations on the surprisingly simple theme given in

This is the vector form of

**Exercise**6 . ) C . Inequalities of Hardy - Hilbert type . Thenext set of inequalities are all variations on the surprisingly simple theme given in

**Exercise**15 , which lends itself to surprisingly manifold ramifications .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero