## Linear Operators: General theory |

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Page 365

Show that the map : f f defined in

manner onto the closed subspace of Le consisting of those F all of whose

negative Fourier coefficients vanish . 55 Using the notations of

54 , show ...

Show that the map : f f defined in

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

negative Fourier coefficients vanish . 55 Using the notations of

**Exercises**53 and54 , show ...

Page 371

( Hint : Generalize the argument of

of the unit disc . ) 88 Show that

function f in H , can be written as a product gh , where g and h are in Hz . ( Hint :

U'se ...

( Hint : Generalize the argument of

**Exercise**85 to apply to zeros on the boundaryof the unit disc . ) 88 Show that

**Exercise**87 is valid even if p = l . 89 Everyfunction f in H , can be written as a product gh , where g and h are in Hz . ( Hint :

U'se ...

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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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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