Linear Operators: General theory |
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Page 2
... sets are the same if and only if they have the same elements , i.e. , A = B if and only if ACB and BCA . The set A ... disjoint if their intersection is void . A family of sets is a disjoint family if every pair of distinct sets in the ...
... sets are the same if and only if they have the same elements , i.e. , A = B if and only if ACB and BCA . The set A ... disjoint if their intersection is void . A family of sets is a disjoint family if every pair of distinct sets in the ...
Page 97
... set S. Then for every E in Σ the total variation of μ on E , denoted by v ( u , E ) , is defined as n v ( μ , E ) = sup Σ μ ( Ε ) , i = 1 where the supremum is taken over all finite sequences { E } of disjoint sets in with E , CE . The set ...
... set S. Then for every E in Σ the total variation of μ on E , denoted by v ( u , E ) , is defined as n v ( μ , E ) = sup Σ μ ( Ε ) , i = 1 where the supremum is taken over all finite sequences { E } of disjoint sets in with E , CE . The set ...
Page 203
... disjoint sets in Σ . By Lemma 18 , there are finite sets л1 , л1⁄2 in A with E1 € Σ , E2 € 22. Thus , if л = л12 , we have E1 , E2 € Σ * Σπ and hence there are disjoint sets A1 , A2 in Σ with 1 π and E1 - = A1 × S E2 = A2XS μ ( E2 ○ E2 ) ...
... disjoint sets in Σ . By Lemma 18 , there are finite sets л1 , л1⁄2 in A with E1 € Σ , E2 € 22. Thus , if л = л12 , we have E1 , E2 € Σ * Σπ and hence there are disjoint sets A1 , A2 in Σ with 1 π and E1 - = A1 × S E2 = A2XS μ ( E2 ○ E2 ) ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ