## Linear Operators: General theory |

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Page 171

23 Suppose that S is a metric space, that E is a cr-field, and that ft is countably

additive and bounded. ... 33 Show that (S, E, ft) can be a finite

23 Suppose that S is a metric space, that E is a cr-field, and that ft is countably

additive and bounded. ... 33 Show that (S, E, ft) can be a finite

**positive measure****space**, and {/a} a uniformly bounded generalized sequence of non-negative ...Page 186

Thus, the field 0 of Lemma 1 is a tx-field and the measure // of Lemma 1 is

countably additive on 0. Consequently, the restriction ... Let (S,E,/i) be the product

of finite

s2 in ...

Thus, the field 0 of Lemma 1 is a tx-field and the measure // of Lemma 1 is

countably additive on 0. Consequently, the restriction ... Let (S,E,/i) be the product

of finite

**positive measure spaces**(Sv Ev fa) and (S2, E2, fa). For each E in E ands2 in ...

Page 302

Let (S, E, ft) he a

LP(S, E, ft), 1 ^ p < oo, is a complete lattice. Proof. It is evidently sufficient to show

that if {/„} is a set of functions in Lj such that 0 ^ fa ąS g0 for some g0 e Lv then ...

Let (S, E, ft) he a

**positive measure space**. Then the real partially ordered spaceLP(S, E, ft), 1 ^ p < oo, is a complete lattice. Proof. It is evidently sufficient to show

that if {/„} is a set of functions in Lj such that 0 ^ fa ąS g0 for some g0 e Lv then ...

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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 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