## Linear Operators: General theory |

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Page 410

1 DEFINITION . A

K . The following lemma is an obvious consequence of Definition 1 . 2 LEMMA .

The intersection of an arbitrary family of

1 DEFINITION . A

**set**K CX is**convex**if x , y eK , and o şasi , imply ax + ( 1 - a ) y eK . The following lemma is an obvious consequence of Definition 1 . 2 LEMMA .

The intersection of an arbitrary family of

**convex**subsets of the linear space X is ...Page 414

( b ) that a bounding point of K is a boundary point of K ; it remains to show that if

the

interior point , and a boundary point q , is a bounding point . Since q , is internal ,

the ...

( b ) that a bounding point of K is a boundary point of K ; it remains to show that if

the

**convex set**K has at least one interior point p , an internal point q1 is aninterior point , and a boundary point q , is a bounding point . Since q , is internal ,

the ...

Page 842

2 – 4 ( 158 – 160 ) Weierstrass theorem on analytic functions , ( 228 ) Convex

combination , V . 2 . 2 ( 414 ) . See also Convex hull ,

) Convex function , definition , VI . 10 . 1 ( 520 ) study of , VI . 10 Convex hull , V . 2

.

2 – 4 ( 158 – 160 ) Weierstrass theorem on analytic functions , ( 228 ) Convex

combination , V . 2 . 2 ( 414 ) . See also Convex hull ,

**Convex set**, Convex space) Convex function , definition , VI . 10 . 1 ( 520 ) study of , VI . 10 Convex hull , V . 2

.

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 |

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