## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 76

Page 88

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

separability

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

Archimedean ...

Page 131

The necessity of the

the positive and negative variations of its real and imaginary parts satisfy the

same ...

The necessity of the

**condition**is obvious . To prove the sufficiency of the**condition**we observe first that a set function a satisfies this**condition**if and only ifthe positive and negative variations of its real and imaginary parts satisfy the

same ...

Page 433

The necessity of the

observe that the

an æ * X * such that x * ( K ) is an unbounded convex set of scalars , and hence ...

The necessity of the

**condition**is immediate from 1 . 5 . 6 . For the sufficiency , weobserve that the

**condition**implies that K is bounded . For otherwise , there existsan æ * X * such that x * ( K ) is an unbounded convex set of scalars , and hence ...

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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