## Linear Operators: General theory |

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Page 239

It consists of ordered n - tuples x = of scalars Q1 , ... , Chen and has the

xz | = { } 10 ; \ " } 1 ' ” . 1 = 1 3 . The space 1 " is the linear space of all ordered n -

tuples [ Q7 , an ] of scalars Oy , ... , Chon with the

It consists of ordered n - tuples x = of scalars Q1 , ... , Chen and has the

**norm**n |xz | = { } 10 ; \ " } 1 ' ” . 1 = 1 3 . The space 1 " is the linear space of all ordered n -

tuples [ Q7 , an ] of scalars Oy , ... , Chon with the

**norm**la | = sup loilo 1Sisn 4 .Page 472

168 ] showed that the

and only if the function x , achieves its maximum at exactly one point . Mazur ( 1 ;

p . 78 - 79 ] proved that the same condition holds in B ( S ) , that the

168 ] showed that the

**norm**in C [ 0 , 1 ] is strongly differentiable at xo e C [ 0 , 1 ] ifand only if the function x , achieves its maximum at exactly one point . Mazur ( 1 ;

p . 78 - 79 ] proved that the same condition holds in B ( S ) , that the

**norm**in Lp ...Page 532

21 Show that the map T defined by ( a ) ( Hardy ) ( Tf ) ( x ) Г. Кузду is a map in L ,

( 0,00 ) of

Tf ) ( x ) = ( TF ) x ) = $ at dy 00 ES is a map in L , ( 0 , ) of

21 Show that the map T defined by ( a ) ( Hardy ) ( Tf ) ( x ) Г. Кузду is a map in L ,

( 0,00 ) of

**norm**p / ( p - 1 ) , p > 1 , ( b ) ( Hilbert , Schur , Hardy , M. Riesz ) f ( y ) (Tf ) ( x ) = ( TF ) x ) = $ at dy 00 ES is a map in L , ( 0 , ) of

**norm**( sin a / p ) -1 , p ...### 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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### 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