## Linear Operators: General theory |

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Page 8

Hence, by Theorem 7, there is a maximal well-ordered

E for if x is in E but not in E0 the ordering ig ... 2 If a family $ of

has the property that A e <? if and only if every finite

f ...

Hence, by Theorem 7, there is a maximal well-ordered

**subset**E0 of E. Now E0 =E for if x is in E but not in E0 the ordering ig ... 2 If a family $ of

**subsets**of a sethas the property that A e <? if and only if every finite

**subset**of A belongs to then <f ...

Page 439

Let K be a

A Q K is said to be an extremal

(l — a)k2, 0 < a < 1, of two points of K is in A only if both A:x and k2 are in A. An ...

Let K be a

**subset**of a real or complex linear vector space X. A non-void**subset**A Q K is said to be an extremal

**subset**of K if a proper convex combination aA;1 +(l — a)k2, 0 < a < 1, of two points of K is in A only if both A:x and k2 are in A. An ...

Page 440

totally ordered subfamily of s/, the non-void set n is a closed extremal

which furnishes a lower bound for $tx. It follows by Zorn's lemma that stf contains

a minimal element A0. Suppose that A0 contains two distinct points p and q.

totally ordered subfamily of s/, the non-void set n is a closed extremal

**subset**of Kwhich furnishes a lower bound for $tx. It follows by Zorn's lemma that stf contains

a minimal element A0. Suppose that A0 contains two distinct points p and q.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero