## Linear Operators, Part 1 |

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Page 464

( 3 ) For every | ET , sup . zek Rf ( x ) < 00 ; moreover , each linear functional Ø on

I which

( 1 ) = f ( x ) , for some X , € K . ( 4 ) If KL , $ < 0 , 0 a limit ordinal , is a monotone ...

( 3 ) For every | ET , sup . zek Rf ( x ) < 00 ; moreover , each linear functional Ø on

I which

**satisfies**inf R } ( x ) = RD ( 1 ) sup R } ( x ) , ter REK ZEK ter also**satisfies**0( 1 ) = f ( x ) , for some X , € K . ( 4 ) If KL , $ < 0 , 0 a limit ordinal , is a monotone ...

Page 557

Consequently , the product R , of all the factors ( a - hilai in R such that di € O ( T )

, still

) Bi , where Bi = min ( die v ( 2 : ) ) ,

Consequently , the product R , of all the factors ( a - hilai in R such that di € O ( T )

, still

**satisfies**R ( T ) = 0 . In the same way , the product R , of all the factors ( 2 - hi) Bi , where Bi = min ( die v ( 2 : ) ) ,

**satisfies**R , ( T ) = 0 . Since any polynomial ...Page 613

... continuous real ( or complex ) function f of the nonnegative real variable t

which

operator valued functions defined on the range t 20 , which

i ) T ( t + ...

... continuous real ( or complex ) function f of the nonnegative real variable t

which

**satisfies**the functional equations f ( 0 ) ... the most general continuousoperator valued functions defined on the range t 20 , which

**satisfy**the equations (i ) T ( t + ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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 obtained 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