## Linear Operators: General theory |

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Page 239

It consists of ordered n - tuples x = [ Q2 , . . . , an ] of scalars dj , . . . , On and has

the

ordered n - tuples x = [ 0 , . . . , Cen ] of scalars dy , . . . , Chon with the

sup ...

It consists of ordered n - tuples x = [ Q2 , . . . , an ] of scalars dj , . . . , On and has

the

**norm**v = { } | 2 | p ] 1 / 0 . 1 = 1 3 . The space I " . is the linear space of allordered n - tuples x = [ 0 , . . . , Cen ] of scalars dy , . . . , Chon with the

**norm**lx | =sup ...

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 € 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 1 x ( a ) ( Hardy ) ( Tf ) ( x ) = - f ( y ) dy x Jo is a

map in L , ( 0 , 00 ) of

Riesz ) poo f ( y ) , ( T } ) ( x ) = 1 Jo c + g is a map in L , ( 0 , 0 ) of

21 Show that the map T defined by 1 x ( a ) ( Hardy ) ( Tf ) ( x ) = - f ( y ) dy x Jo is a

map in L , ( 0 , 00 ) of

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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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 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