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 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 ( ○ E1 ) = Σ μ ( E ...
... extended . 1 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 ( ○ E1 ) = Σ μ ( E ...
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 . μ PROOF . If u 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 . μ PROOF . If u is a regular additive complex or extended ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ