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Page 77
... converges . 48 Show that Σαπ n = 1 = ∞ Σαπ / sin nh \ nh lim 2a . ( sin nh h → 0 n = 1 whenever k > 1 and the series on the left converges . m n = 0 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex ...
... converges . 48 Show that Σαπ n = 1 = ∞ Σαπ / sin nh \ nh lim 2a . ( sin nh h → 0 n = 1 whenever k > 1 and the series on the left converges . m n = 0 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex ...
Page 145
... converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( u , E ) < ɛ and such that { n } converges uniformly to ƒ on S - E . It is clear that u ...
... converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( u , E ) < ɛ and such that { n } converges uniformly to ƒ on S - E . It is clear that u ...
Page 595
... converge for 0 ≤m < a ( 2 ) , and if lim , fn ( 20 ) 0 , then f ( T ) } converges in the weak operator topology . Moreover , n * = ® f ( T ) X → { xx € X , f ( T ) x = 0 } . = { x | x € X , f ( T ) x = 0 } , PROOF . Let 1 = f ( T ) X ...
... converge for 0 ≤m < a ( 2 ) , and if lim , fn ( 20 ) 0 , then f ( T ) } converges in the weak operator topology . Moreover , n * = ® f ( T ) X → { xx € X , f ( T ) x = 0 } . = { x | x € X , f ( T ) x = 0 } , PROOF . Let 1 = f ( T ) X ...
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