## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 82

Page 239

It consists of ordered n - tuples x = [ ( y , . . . , am ] of scalars a , . . . , On and has

the

ordered n - tuples x = [ 1 , . . . , An ] of scalars Oy , . . . , On with the

laila ...

It consists of ordered n - tuples x = [ ( y , . . . , am ] of scalars a , . . . , On and has

the

**norm**1x1 = { } 10 . [ P ] 1 / 0 . 3 . The space Ime is the linear space of allordered n - tuples x = [ 1 , . . . , An ] of scalars Oy , . . . , On with the

**norm**\ w ] = suplaila ...

Page 240

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

the

sequences x = { an } for which the series - Olon is convergent . The

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

the

**norm**1x1 = sup / Oil i = 1 is finite . 11 . The space cs is the linear space of allsequences x = { an } for which the series - Olon is convergent . The

**norm**is n Q ...Page 532

21 Show that the map T defined by ( a ) ( Hardy ) ( T / ) ( x ) = - f ( y ) dy X JO is a

map in Ly ( 0 , 0 ) of

Riesz ) ( T } ) ( x ) = * t ( y ) - dy | Jo ac + g is a map in L , ( 0 , 0 ) of

p ) ...

21 Show that the map T defined by ( a ) ( Hardy ) ( T / ) ( x ) = - f ( y ) dy X JO is a

map in Ly ( 0 , 0 ) of

**norm**p / ( p - 1 ) , p > 1 , ( b ) ( Hilbert , Schur , Hardy , M .Riesz ) ( T } ) ( x ) = * t ( y ) - dy | Jo ac + g is a map in L , ( 0 , 0 ) of

**norm**n ( sin n /p ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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 |

### Common terms and phrases

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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