## Linear Operators, Part 1 |

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Page 83

That such results are valid in metric groups was

Banach [ 1 ; Chap . 1 ] and Kuratowski [ 1 ] ) . Conditions of this nature are

extended to polynomial operators by Mazur and Orlicz [ 2 ] . It is sometimes useful

to ...

That such results are valid in metric groups was

**shown**by Banach ( 7 ] ( see alsoBanach [ 1 ; Chap . 1 ] and Kuratowski [ 1 ] ) . Conditions of this nature are

extended to polynomial operators by Mazur and Orlicz [ 2 ] . It is sometimes useful

to ...

Page 271

Hence we have

} < , ISIST which proves the quasi - uniform convergence of f ( sn ) to f ( so ) . Thus

( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) . Let F. { t1 f2 , ... } be a ...

Hence we have

**shown**that there exists nī , No 2 no such that min \ | ( snc ) - ( so )} < , ISIST which proves the quasi - uniform convergence of f ( sn ) to f ( so ) . Thus

( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) . Let F. { t1 f2 , ... } be a ...

Page 553

841 ] has

seen in Theorems 7.4 and 8.12 that in the spaces C and Ll , FCCC W CP .

Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and P coincide . This is

not ...

841 ] has

**shown**that in Hilbert space C is a maximal two - sided ideal . We haveseen in Theorems 7.4 and 8.12 that in the spaces C and Ll , FCCC W CP .

Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and P coincide . This is

not ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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 |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero