## Linear Operators, Part 1 |

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Results 1-3 of 79

Page 88

x = p ( x ) + pl - x ) , then this condition is

separability condition cannot be dropped . Ingleton [ 1 ] has given conditions for

the Hahn - Banach theorem to hold when the field of scalars is non -

Archimedean ...

x = p ( x ) + pl - x ) , then this condition is

**sufficient**. Bonsall [ 1 ] showed that theseparability condition cannot be dropped . Ingleton [ 1 ] has given conditions for

the Hahn - Banach theorem to hold when the field of scalars is non -

Archimedean ...

Page 383

30 – 36 ] independently demonstrated that uniform convergence is

assure the continuity of the limit function . ( It is interesting that Weierstrass [ 1 ; p .

67 , 70 ] had employed this notion of convergence in some unpublished ...

30 – 36 ] independently demonstrated that uniform convergence is

**sufficient**toassure the continuity of the limit function . ( It is interesting that Weierstrass [ 1 ; p .

67 , 70 ] had employed this notion of convergence in some unpublished ...

Page 385

ing conditions that are

entire space . Wermer [ 6 , 9 ] has given

generate C ( S ) when S is the unit circle . If f is one - to - one , or the function ( 2 )

...

ing conditions that are

**sufficient**to insure that a class of functions generates theentire space . Wermer [ 6 , 9 ] has given

**sufficient**conditions that two functionsgenerate C ( S ) when S is the unit circle . If f is one - to - one , or the function ( 2 )

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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