Linear Operators: General theory |
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Page 11
... LEMMA . If ẞ is a family of subsets of X , and if x is the family of all unions of subfamilies of ẞ , then is a topology if and only if ( i ) for every pair U , V e ẞ and x e UV there is a We ẞ , such that a WCUV , and ( ii ) X = Uẞ . 8 ...
... LEMMA . If ẞ is a family of subsets of X , and if x is the family of all unions of subfamilies of ẞ , then is a topology if and only if ( i ) for every pair U , V e ẞ and x e UV there is a We ẞ , such that a WCUV , and ( ii ) X = Uẞ . 8 ...
Page 699
... lemma . Q.E.D. We shall now state and prove the lemma referred to as CP . For technical reasons occurring later the following lemma is stated for what might be called a positive sub - semi - group rather than for a positive semi - group ...
... lemma . Q.E.D. We shall now state and prove the lemma referred to as CP . For technical reasons occurring later the following lemma is stated for what might be called a positive sub - semi - group rather than for a positive semi - group ...
Page 700
... lemma is known to be true for an integer k it is also true for any integer mk . Thus , to prove the lemma , it will suffice to show that it is true if k is even . If k = 1 the lemma has al- ready been proved ( Lemma 6 ) . We shall thus ...
... lemma is known to be true for an integer k it is also true for any integer mk . Thus , to prove the lemma , it will suffice to show that it is true if k is even . If k = 1 the lemma has al- ready been proved ( Lemma 6 ) . We shall thus ...
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