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Page 77
... converges . 48 Show that 00 Σαπ - n = 1 ∞ nh \ lim Σαπ ( sin nk ) * h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex numbers , and let ...
... converges . 48 Show that 00 Σαπ - n = 1 ∞ nh \ lim Σαπ ( sin nk ) * h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex numbers , and let ...
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 ( μ , E ) < ɛ and such that { n } converges uniformly to f 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 ( μ , E ) < ɛ and such that { n } converges uniformly to f on S - E . It is clear that u ...
Page 595
... converge for 0 ≤ma ( 2 ) , and if limo fn ( 20 ) 0 , then { f ( T ) } converges in the weak operator topology . Moreover , n X = = f ( T ) X ® { x | x € X , f ( T ) x = 0 } . PROOF . Let X1 = f ( T ) X , X2 = { x | x € X , f ( T ) x ...
... converge for 0 ≤ma ( 2 ) , and if limo fn ( 20 ) 0 , then { f ( T ) } converges in the weak operator topology . Moreover , n X = = f ( T ) X ® { x | x € X , f ( T ) x = 0 } . PROOF . Let X1 = f ( T ) X , X2 = { x | x € X , 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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ