## Linear Operators: General theory |

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Page 88

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

separability

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

Archimedean .

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 -

Archimedean .

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 2satisfies 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 ais ...

Page 433

The necessity of the

observe that the

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

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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