## Linear Operators: General theory |

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Page 410

The intersection of an arbitrary family of convex subsets of the linear space X is

convex. As examples of convex subsets of X, we note the subspaces of BE, and

the subsets of X consisting of one point. 3 Lemma. Let be points in the

...

The intersection of an arbitrary family of convex subsets of the linear space X is

convex. As examples of convex subsets of X, we note the subspaces of BE, and

the subsets of X consisting of one point. 3 Lemma. Let be points in the

**convex set**...

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 q1 is internal, the ...

(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 q1 is internal, the ...

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 | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact