## Linear Operators: General theory |

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Page 60

Statement ( iv ) clearly

i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

= 0 , la si Tx ] = [ v TC . ) SM \ x } . This shows that ( iii )

Statement ( iv ) clearly

**implies**the continuity of T at 0 ; so ( iv )**implies**( ii ) . This (i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

= 0 , la si Tx ] = [ v TC . ) SM \ x } . This shows that ( iii )

**implies**( iv ) . It is obvious ...Page 280

That ( 1 )

14 to show that condition ( 3 ) of that theorem

follows that S may be embedded as a dense subset of a compact Hausdorff

space ...

That ( 1 )

**implies**( 2 ) can be proved in a manner similar to that used in Theorem14 to show that condition ( 3 ) of that theorem

**implies**( 4 ) . From Corollary 19 itfollows that S may be embedded as a dense subset of a compact Hausdorff

space ...

Page 394

A real B - space is said to be an abstract L - space if it is a vector lattice in which x

20 and y 20

norm which satisfies ( 1 ) and also the condition ( m ) XA Y = 0

...

A real B - space is said to be an abstract L - space if it is a vector lattice in which x

20 and y 20

**imply**1x + y = ( x + yl . ... space can be provided with an equivalentnorm which satisfies ( 1 ) and also the condition ( m ) XA Y = 0

**implies**( x + y = | x...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero