## Linear Operators: General theory |

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Page 7

Indeed , it will be

where x s'y , whenever x and y both belong to some subset E , E En , and x Soy in

the ordering Fo of that En . It is clear that if ( UE , S ' ) belongs to & it is an upper ...

Indeed , it will be

**seen**that this upper bound may be defined as ( UE , S ' ) ,where x s'y , whenever x and y both belong to some subset E , E En , and x Soy in

the ordering Fo of that En . It is clear that if ( UE , S ' ) belongs to & it is an upper ...

Page 24

Now it is readily

the uniform continuity of g follows . Finally , the uniqueness of g is obvious . Q . E .

D . In general , a continuous function is not uniformly continuous , but on a ...

Now it is readily

**seen**that g ( x , x ' ) < d implies e ( g ( x ) , g ( x ' ) ) < € ; from thisthe uniform continuity of g follows . Finally , the uniqueness of g is obvious . Q . E .

D . In general , a continuous function is not uniformly continuous , but on a ...

Page 254

Now since UE U there is a finite chain of the form given in ( * ) above in which

successive vectors have non - zero scalar products . Thus by forming the chain

Up , U , • , Ugis Vg , it is

Since ...

Now since UE U there is a finite chain of the form given in ( * ) above in which

successive vectors have non - zero scalar products . Thus by forming the chain

Up , U , • , Ugis Vg , it is

**seen**that vp is equivalent to vge and thus that is in V.Since ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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