## Linear Operators: General theory |

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Results 1-3 of 91

Page 8

Nelson Dunford, Jacob T. Schwartz. Hence, by Theorem 7, there is a maximal

well-ordered

S0 in E0 may be extended to the set EQ u {x} by defining y g0 x for y e E0. Q.E.D.

3.

Nelson Dunford, Jacob T. Schwartz. 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 :S0 in E0 may be extended to the set EQ u {x} by defining y g0 x for y e E0. Q.E.D.

3.

Page 439

Let Ar be a

A Q K is said to be an extremal

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

Let Ar 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 if is in A only if both /cx and k2 are in A. An ...

Page 440

totally ordered subfamily of s/. the non-void set O s#x is a closed extremal

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

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

and q.

totally ordered subfamily of s/. the non-void set O s#x is a closed extremal

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

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 | 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