## Linear Operators: General theory |

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Page 88

1x1 = p ( x ) + p ( - x ) , then this

the separability

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

1x1 = p ( x ) + p ( - x ) , then this

**condition**is sufficient . Bonsall [ 1 ] showed thatthe separability

**condition**cannot be dropped . Ingleton [ 1 ] has given**conditions**for the Hahn - Banach theorem to hold when the field of scalars is non ...

Page 131

To prove the sufficiency of the

satisfies this

and imaginary parts satisfy the same

is ...

To prove the sufficiency of the

**condition**we observe first that a set function àsatisfies this

**condition**if and only if the positive and negative variations of its realand imaginary parts satisfy the same

**condition**. Thus , it may be assumed that 2is ...

Page 487

6 , the

subset of C ( S * ) . 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 ( S * ) . 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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 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