## Linear Operators: General theory |

### From inside the book

Results 1-3 of 86

Page 8

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

E for if a: is in £ but not in E0 the ordering 5S0 in E0 may be extended to the set

E0 (j {x} by defining y :g0 x for y e E0. Q.E.D. 3. Exercises 1 If (E, :S) 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 a: is in £ but not in E0 the ordering 5S0 in E0 may be extended to the set

E0 (j {x} by defining y :g0 x for y e E0. Q.E.D. 3. Exercises 1 If (E, :S) is a p.o.s.

with ...

Page 440

totally ordered subfamily of s/. the non-void set n s/1 is a closed extremal

of A' which furnishes a lower bound for s/y It follows by Zona's lemma that &/

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 s/1 is a closed extremal

**subset**of A' which furnishes a lower bound for s/y It follows by Zona's lemma that &/

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

and q.

Page 459

12 Let X be a B-space, and let K be a weakly compact convex

that K has a continuous tangent functional at each point of a dense

boundary. 13 Let X be a J3-space, and let K* be a bounded X-closed convex ...

12 Let X be a B-space, and let K be a weakly compact convex

**subset**of X. Showthat K has a continuous tangent functional at each point of a dense

**subset**of itsboundary. 13 Let X be a J3-space, and let K* be a bounded X-closed convex ...

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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 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