## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 71

Page 410

A set K CX is

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

intersection of an arbitrary family of

A set K CX is

**convex**if x , y eK , and 0 Sa 31 , imply ax + ( 1 - a ) y e K. Thefollowing 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**convex**.Page 418

If Ky , K , are disjoint closed ,

topological space X , and if K , is compact , then some non - zero continuous

linear functional on X separates K , and Ką . 12 COROLLARY . If K is a closed

If Ky , K , are disjoint closed ,

**convex**subsets of a locally**convex**lineartopological space X , and if K , is compact , then some non - zero continuous

linear functional on X separates K , and Ką . 12 COROLLARY . If K is a closed

**convex**subset of ...Page 842

See also

VI.10.1 ( 520 ) study of , VI.10

definition , V.1.1 ( 410 ) study of , V.1-2

See also

**Convex**hull ,**Convex**set ,**Convex**space )**Convex**function , definition ,VI.10.1 ( 520 ) study of , VI.10

**Convex**hull , V.2.2 ( 414 )**Convex**set , II.4.1 ( 70 )definition , V.1.1 ( 410 ) study of , V.1-2

**Convex**space , locally , V.2.9 ( 417 ) ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero