## Linear Operators, Part 1 |

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Page 88

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

separability

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

x1 = 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 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 ...

Page 487

in S. Hence , by IV.6.6 , the

an equicontinuous 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

in S. Hence , by IV.6.6 , the

**condition**is equivalent to the statement that T ( S ) isan equicontinuous 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

**condition**is satisfied .### 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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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