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 λ defined above we have functions f € 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 λ defined above we have functions f € L1 ...
Page 341
... respect to which every in K is continuous . ( iii ) lim Uλ = uniformly with respect to λ € K. a 20 Let Σ = { E } be a countable field of subsets of a set S , and let 1 be the o - field generated by E. Let μ be a non - negative finite ...
... respect to which every in K is continuous . ( iii ) lim Uλ = uniformly with respect to λ € K. a 20 Let Σ = { E } be a countable field of subsets of a set S , and let 1 be the o - field generated by E. Let μ be a non - negative finite ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ