## 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 E , and x Soy in

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

bound ...

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 E , and x Soy in

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

bound ...

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 Uq ' E 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

VR , Uq . . . , Ug ' s Vg , it is

...

Now since Uq ' E 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

VR , Uq . . . , Ug ' s Vg , it is

**seen**that vg is equivalent to vg , and thus that Vg is in...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

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