Linear Operators: General theory |
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Page 7
... belong to some subset E。€ 80 , and x ≤oy in the ordering of that E 。. It is clear that if ( UE , ≤ ' ) belongs to & it is an upper bound for &。. It will now be shown that it is well - ordered and hence belongs to & . The statement ...
... belong to some subset E。€ 80 , and x ≤oy in the ordering of that E 。. It is clear that if ( UE , ≤ ' ) belongs to & it is an upper bound for &。. It will now be shown that it is well - ordered and hence belongs to & . The statement ...
Page 184
... belong to A1 . It follows that Ø is closed under intersection and complementation and contains S. Thus is a field of sets . Moreover , each set of & belongs to D and hence . If the set function u is defined for Fe by the formula μ ( F ) ...
... belong to A1 . It follows that Ø is closed under intersection and complementation and contains S. Thus is a field of sets . Moreover , each set of & belongs to D and hence . If the set function u is defined for Fe by the formula μ ( F ) ...
Page 659
... belongs to a suitable class of functions on S. The reader will observe that the family of such operators certainly includes those of the form f ( · ) → ƒ ( þ ( · ) ) . It will be seen that the discussion which we give in the sequel ...
... belongs to a suitable class of functions on S. The reader will observe that the family of such operators certainly includes those of the form f ( · ) → ƒ ( þ ( · ) ) . It will be seen that the discussion which we give in the sequel ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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 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