## Linear Operators, Part 1 |

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Page 410

1 Definition. A set K Q X is

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

intersection of an arbitrary family of

1 Definition. A set K Q X is

**convex**ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.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**convex**.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 s 1 , 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 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 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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