## Linear Operators: General theory |

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Page 410

A set K Q X is

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

intersection of an arbitrary family of

A set K Q X is

**convex**iix,ye K, and 0 52 a 52 1, imply ax-\-(\—a)y eK. 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 410

A set K CX is

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

intersection of an arbitrary family of

A set K CX is

**convex**if x , y e K , and 0 sa si , imply ax + ( 1 - a ) y € K. Thefollowing lemma is an obvious consequence of Definition 1 . LEMMA . The

intersection of an arbitrary family of

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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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