## Linear Operators: General theory |

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Page 410

A

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**ifx,ye K, and 0 5g a ^ 1, imply ax + (l — 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 414

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

the

point, and a boundary point q2 is a bounding point. Since qt is internal, the vector

...

(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 an interiorpoint, and a boundary point q2 is a bounding point. Since qt is internal, the vector

...

Page 842

6 (122), III.6.15 (150), III.9.45 (173), IV.10.9 (325) Vitali-Hahn-Saks theorem for

measures, III.7.2-4 (158-100) Weierstrass theorem on analytic functions, (228)

Convex combination, V.2.2 (414). See also Convex hull.

space) ...

6 (122), III.6.15 (150), III.9.45 (173), IV.10.9 (325) Vitali-Hahn-Saks theorem for

measures, III.7.2-4 (158-100) Weierstrass theorem on analytic functions, (228)

Convex combination, V.2.2 (414). See also Convex hull.

**Convex set**, Convexspace) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero