## Linear Operators: General theory |

### From inside the book

Results 1-3 of 87

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 ^0 in E0 may be extended to the set E0

u {x} by defining y ;£„ x for y e £0. Q.E.D. 3. Exercises 1 If (E, :£) is a p.o.s. with ...

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 ^0 in E0 may be extended to the set E0

u {x} by defining y ;£„ x for y e £0. Q.E.D. 3. Exercises 1 If (E, :£) is a p.o.s. with ...

Page 439

Let A' be a

A Q A is said to be an extremal

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

Let A' be a

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

**subset**of A if a proper convex combination afc1 +(l — a)kv 0 < a < 1, of two points of A is in A only if both fcj and k2 are in A. An ...

Page 440

totally ordered subfamily of sJ '. the non-void set ru/1 is a closed extremal

of A' which furnishes a lower bound for s/v It follows by Zorn's lemma that

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

and q.

totally ordered subfamily of sJ '. the non-void set ru/1 is a closed extremal

**subset**of A' which furnishes a lower bound for s/v It follows by Zorn's lemma that

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 | |

44 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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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