Linear Operators: General theory |
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Page 96
... variation of μ . The set function v ( u ) is defined so as to be equal to u if u itself is non - negative , to be additive if u is additive , and to be bounded if u is bounded . The 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
... variation of μ . The set function v ( u ) is defined so as to be equal to u if u itself is non - negative , to be additive if u is additive , and to be bounded if u is bounded . The 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
Page 97
... additive set function u is important because it dominates u in the sense that v ( μ , E ) ≥ │μ ( E ) for E € Σ ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
... additive set function u is important because it dominates u in the sense that v ( μ , E ) ≥ │μ ( E ) for E € Σ ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
Page 126
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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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