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 . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ ...
... field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a field Σ ...
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 € Σ , and that if Σ is a σ - field , then Σ ( E ) is a σ - field . ( E ) is called the restriction of Σ to E. If Σ is a field , E € Σ1 , and Σ is the o - field ...
... field of subsets of E , and that Σ ( E ) is the family of all sets AE , A € Σ , and that if Σ is a σ - field , then Σ ( E ) is a σ - field . ( E ) is called the restriction of Σ to E. If Σ is a field , E € Σ1 , and Σ is the o - field ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ