Linear Operators: General theory |
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Page 95
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function μ , the resulting integral v may be a vector valued set function . It is ...
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function μ , the resulting integral v may be a vector valued set function . It is ...
Page 126
... 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 ; ) i = 1 Σ whenever E1 , E2 , ... are ...
... 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 ; ) i = 1 Σ whenever E1 , E2 , ... are ...
Page 323
... valued measurable function ƒ is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u - almost everywhere ; ( ii ) the ... valued and u IV.10.7 323 VECTOR VALUED MEASURES.
... valued measurable function ƒ is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u - almost everywhere ; ( ii ) the ... valued and u IV.10.7 323 VECTOR VALUED MEASURES.
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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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