Linear Operators: General theory |
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Page 95
... real function , complex function , etc. , where the adjectives real and ... valued function and ƒ a vector ( or scalar ) valued function . Thus , even ... extended real number system ( which is not a vector space ) but , since ∞ + ...
... real function , complex function , etc. , where the adjectives real and ... valued function and ƒ a vector ( or scalar ) valued function . Thus , even ... extended real number system ( which is not a vector space ) but , since ∞ + ...
Page 126
... extended . 1 DEFINITION . Let μ 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 μ is said to be countably additive if 00 μ ( UE ; ) i = 1 = 00 Σμ ( Ε ...
... extended . 1 DEFINITION . Let μ 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 μ is said to be countably additive if 00 μ ( UE ; ) i = 1 = 00 Σμ ( Ε ...
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 u is a regular additive complex or extended real ...
... 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 u is a regular additive complex or extended real ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ