Linear Operators: General theory |
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Page 97
... additive set function u is important because it dominates μ in the sense that v ( μ , E ) ≥ │μ ( E ) | for E 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 μ in the sense that v ( μ , E ) ≥ │μ ( E ) | for E 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
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued ...
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued ...
Page 233
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ