## 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 430

It is readily seen that ( iii )

decidedly non - trivial ; we will complete the proof by showing first that ( ii )

Let { Xn } ...

It is readily seen that ( iii )

**implies**( ii ) . The other implications we desire aredecidedly non - trivial ; we will complete the proof by showing first that ( ii )

**implies**( i ) , and then that ( i )**implies**( iii ) . Proof that condition ( ii )**implies**( i ) .Let { Xn } ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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

algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero