Linear Operators: General theory |
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Page 126
... field is a field 2 of subsets of a set S with the property that Ee whenever E , e , n = 1 , 2 , .. .. In other words , a σ - field is a field which is closed under the operation of forming denumerable unions . 3 DEFINITION . A triple ...
... field is a field 2 of subsets of a set S with the property that Ee whenever E , e , n = 1 , 2 , .. .. In other words , a σ - field is a field which is closed under the operation of forming denumerable unions . 3 DEFINITION . A triple ...
Page 136
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family 7. The intersection of all fields ...
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family 7. The intersection of all fields ...
Page 166
... field of subsets of E , and that Σ ( E ) is the family of all sets AE , A e E , and that if Σ is a σ - field , then ( E ) is a σ - field . Σ ( Ε ) is called the restriction of E to E. If I is a field , E € Σ1 , and E is the o - field ...
... field of subsets of E , and that Σ ( E ) is the family of all sets AE , A e E , and that if Σ is a σ - field , then ( E ) is a σ - field . Σ ( Ε ) is called the restriction of E to E. If I is a field , E € Σ1 , and E is the o - field ...
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