## Linear Operators: General theory |

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Page 410

A

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

an arbitrary family of

A

**set**K Q X is**convex**if x, y e K, and 0 a ^ 1, imply ax + (1— a)y e K. The followinglemma 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

convex ...

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

convex ...

Page 842

See also Convex hull,

10.1 (520) study of, VI.10 Convex hull, V.2.2 (414)

definition, V.l.l (410) study of, V.l-2 Convex space, locally, V.2.9 (417), (471 )

strictly, ...

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.l.l (410) study of, V.l-2 Convex space, locally, V.2.9 (417), (471 )

strictly, ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero