## Linear Operators: General theory |

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Results 1-3 of 79

Page 88

1x = p ( x ) + p ( - x ) , then this

separability

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

1x = p ( x ) + p ( - 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 ...

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 2 satisfies this**condition**if and only ifthe positive and negative variations of its real and imaginary parts satisfy the

same ...

Page 487

6 , the

subset of C ( $ * ) . It follows from Theorem IV . 6 . 7 , that T ( S ) is conditionally

compact in the metric of Y if and only if the

6 , the

**condition**is equivalent to the statement that T ( S ) is an equicontinuoussubset of C ( $ * ) . It follows from Theorem IV . 6 . 7 , that T ( S ) is conditionally

compact in the metric of Y if and only if the

**condition**is satisfied . Q . E . D . 6 .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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