Linear Operators: General theory |
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Page 195
Nelson Dunford, Jacob T. Schwartz. with respect to one variable and then with respect to the other , or vice versa . Indeed , according to Tonelli's theorem , both these inte- grals are equal to the integral of ƒ with respect to the ...
Nelson Dunford, Jacob T. Schwartz. with respect to one variable and then with respect to the other , or vice versa . Indeed , according to Tonelli's theorem , both these inte- grals are equal to the integral of ƒ with respect to the ...
Page 306
... respect to ƒ in K ' and hence uniform with respect to μ in K. Conversely , suppose that the set KC ca ( S , Σ ) satisfies the two conditions and let μn e K , n = 1 , 2 , . . .. Using the measure 2 defined above we have functions fn e L1 ...
... respect to ƒ in K ' and hence uniform with respect to μ in K. Conversely , suppose that the set KC ca ( S , Σ ) satisfies the two conditions and let μn e K , n = 1 , 2 , . . .. Using the measure 2 defined above we have functions fn e L1 ...
Page 341
... respect to which μ every in K is continuous . ( iii ) lim Uλ = uniformly with respect to λe K. λ λε Ul a = 20 Let Σ { E } be a countable field of subsets of a set S , and let be the o - field generated by E. Let u be a non - negative ...
... respect to which μ every in K is continuous . ( iii ) lim Uλ = uniformly with respect to λe K. λ λε Ul a = 20 Let Σ { E } be a countable field of subsets of a set S , and let be the o - field generated by E. Let u be a non - negative ...
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