Linear Operators: General theory |
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Page 100
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
Page 150
... everywhere . 13 18 COROLLARY . ( a ) If ( S , E , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space ...
... everywhere . 13 18 COROLLARY . ( a ) If ( S , E , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space ...
Page 676
... everywhere for every h in L ,. Thus , by Theorem IV.11.2 , the sequence A ( T , n ) h converges almost everywhere for every h in L „ . Since L dense in L1 we may apply Lemma 5 and Theorem IV.11.2 again to see that the sequence A ( T , n ) ...
... everywhere for every h in L ,. Thus , by Theorem IV.11.2 , the sequence A ( T , n ) h converges almost everywhere for every h in L „ . Since L dense in L1 we may apply Lemma 5 and Theorem IV.11.2 again to see that the sequence A ( T , n ) ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ