## Linear Operators: General theory |

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Page 287

Thus | * = Igla The mapping w * →g is then a one - to - one isometric map of L *

into Lg . It is

€ L * satisfying ( i ) , so that the mapping x * 4 g is a one - to - one isometric map

of ...

Thus | * = Igla The mapping w * →g is then a one - to - one isometric map of L *

into Lg . It is

**evident**from the Hölder inequality that any g € La determines an æ *€ L * satisfying ( i ) , so that the mapping x * 4 g is a one - to - one isometric map

of ...

Page 299

... it follows that all the functions g1 , . . . , gy vanish outside some sufficiently large

interval [ - A0 , + 4 . ) , so that Så + S * \ / ( y ) | Pdy = $ * + s4116 ) – 8 ( y ) | Pdy =

lf - gil " SEP for A 2 A0 , proving ( b ) . To prove ( a ) we note first that it is

... it follows that all the functions g1 , . . . , gy vanish outside some sufficiently large

interval [ - A0 , + 4 . ) , so that Så + S * \ / ( y ) | Pdy = $ * + s4116 ) – 8 ( y ) | Pdy =

lf - gil " SEP for A 2 A0 , proving ( b ) . To prove ( a ) we note first that it is

**evident**...Page 337

It is then

isomorphic with the closed subspace BV ( I ) of all / e BV ( I ) such that f ( a + ) = 0

. If N is the one - dimensional space of constant functions , it is

= BV ...

It is then

**evident**that vlug , I ) = v ( 1 , 1 ) . Thus , ba ( S , E ) is isometricallyisomorphic with the closed subspace BV ( I ) of all / e BV ( I ) such that f ( a + ) = 0

. If N is the one - dimensional space of constant functions , it is

**evident**that BV ( I )= BV ...

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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