Linear Operators: General theory |
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Page 126
... field of subsets of a set . In this case the results of the preceding sections can be considerably extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ of ...
... field of subsets of a set . In this case the results of the preceding sections can be considerably extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ of ...
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 τ . 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 τ . The intersection of all fields ...
Page 166
... field of subsets of E , and that of all sets AE , A e E , and that if is a σ - field , then Σ ( Ε ) is called the restriction of E to E. If I is a field , Ε € Σ1 , and I is the o - field generated by 2 , then we can easily show that ( E ) ...
... field of subsets of E , and that of all sets AE , A e E , and that if is a σ - field , then Σ ( Ε ) is called the restriction of E to E. If I is a field , Ε € Σ1 , and I is the o - field generated by 2 , then we can easily show that ( E ) ...
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