Linear Operators: General theory |
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Page 137
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that | μ ( C ) | < ɛ for every C in Σ with CCG - F . € For a complex or extended ...
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that | μ ( C ) | < ɛ for every C in Σ with CCG - F . € For a complex or extended ...
Page 168
... field , and clearly the smallest field containing . An elementary induction shows that C , n = 1 , 2 , . . . , is countable and thus is a countable field . Q.E.D. 5 LEMMA . Let ( S , E , u ) be a positive measure space and G a separable ...
... field , and clearly the smallest field containing . An elementary induction shows that C , n = 1 , 2 , . . . , is countable and thus is a countable field . Q.E.D. 5 LEMMA . Let ( S , E , u ) be a positive measure space and G a separable ...
Page 634
... G ) ( t ) = [ ' F ( t − 8 ) G ( s ) ds . In the next two lemmas are proved certain basic properties of the ... field of Borel subsets of the line , p ( E ) is in the Lebesgue extension of the o - field of Borel subsets of the plane ...
... G ) ( t ) = [ ' F ( t − 8 ) G ( s ) ds . In the next two lemmas are proved certain basic properties of the ... field of Borel subsets of the line , p ( E ) is in the Lebesgue extension of the o - field of Borel subsets of the plane ...
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