## Linear Operators, Part 1 |

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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 şi , 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 Kg . 12 COROLLARY . I / 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 Kg . 12 COROLLARY . I / K is a closed

**convex**subset ...Page 461

and an arbitrary

Theorem 2.8 ) . He also proved that a

topology of the normed linear space X , can be separated from an arbitrary

closed ...

and an arbitrary

**convex**set is possible , provided they are disjoint ( compareTheorem 2.8 ) . He also proved that a

**convex**set K which is compact in the X *topology of the normed linear space X , can be separated from an arbitrary

closed ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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