Linear Operators: General theory |
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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
... o in μ - measure , it follows from Definition 2.17 that | ƒ ( · ) | ” is μ - integrable and thus that ƒ € L2 ( μ1 ) ... 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 ) ...
... o in μ - measure , it follows from Definition 2.17 that | ƒ ( · ) | ” is μ - integrable and thus that ƒ € L2 ( μ1 ) ... 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 ) ...
Page 168
... of L ( S , E , μ , X ) , where 1 ≤ p < ∞o . Then there is a set S1 in Σ , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X1 of such that the restriction μ1 of μ to Σ1 has the following properties : ( i ) the measure ...
... of L ( S , E , μ , X ) , where 1 ≤ p < ∞o . Then there is a set S1 in Σ , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X1 of such that the restriction μ1 of μ to Σ1 has the following properties : ( i ) the measure ...
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