## Linear Operators, Part 1 |

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Page 239

It consists of ordered n - tuples x = of scalars old , ... , Chon and has the

= { 104 | P } 1 p . Žia 1 = 1 r = 3. The space I " is the linear space of all ordered n -

tuples [ Q , ... , an ] of scalars dq , ... , Own with the

It consists of ordered n - tuples x = of scalars old , ... , Chon and has the

**norm**| x |= { 104 | P } 1 p . Žia 1 = 1 r = 3. The space I " is the linear space of all ordered n -

tuples [ Q , ... , an ] of scalars dq , ... , Own with the

**norm**= suplovila 1Sisn 4 .Page 240

The space bs is the linear space of all sequences x = scalars for which the

any of 72 | x | = sup 1 ail n i = 1 is finite . 11. The space cs is the linear space of all

sequences x = for which the series en Oon is convergent . The

The space bs is the linear space of all sequences x = scalars for which the

**norm**{any of 72 | x | = sup 1 ail n i = 1 is finite . 11. The space cs is the linear space of all

sequences x = for which the series en Oon is convergent . The

**norm**is { an } ...Page 532

21 Show that the map T defined by ( a ) ( Hardy ) ( Tf ) ( x ) = f ( y ) dy ES is a map

in L , ( 0 , 00 ) of

( y ) ( Tf ) ( x ) = dy 0 x + y is a map in L , ( 0 , 0 ) of

21 Show that the map T defined by ( a ) ( Hardy ) ( Tf ) ( x ) = f ( y ) dy ES is a map

in L , ( 0 , 00 ) of

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

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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero