Linear Operators: General theory |
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Page 95
... extended real number system , in which case its range contains at most one of the improper values + and -∞ . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A ...
... extended real number system , in which case its range contains at most one of the improper values + and -∞ . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A ...
Page 126
... extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 ∞ μ ( UE ) = Σμ ( Ε . ) i = 1 i ...
... extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 ∞ μ ( UE ) = Σμ ( Ε . ) i = 1 i ...
Page 137
... extended real valued set function on a field is regular . Moreover , the positive and negative variations of a bounded regular real valued additive set function are also regular . PROOF . If μ is a regular additive complex or extended ...
... extended real valued set function on a field is regular . Moreover , the positive and negative variations of a bounded regular real valued additive set function are also regular . PROOF . If μ is a regular additive complex or extended ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ