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Page 77
... converges . 48 Show that sin nh \ Σαπ = lim Σαπ nh n = 1 h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a { a } and b quences of complex numbers , and let cm = n = ' m { b } be two se- Σn ...
... converges . 48 Show that sin nh \ Σαπ = lim Σαπ nh n = 1 h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a { a } and b quences of complex numbers , and let cm = n = ' m { b } be two se- Σn ...
Page 145
... converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each ɛ > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . It is clear that μ ...
... converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each ɛ > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . It is clear that μ ...
Page 595
... converge for 0 ≤m < x ( 2 ) , and if lim , fn ( 20 ) 0 , then { f ( T ) } converges in the weak operator topology . Moreover , = x f ( T ) X { xx X , ƒ ( T ) x = 0 } . PROOF . Let 1 = ƒ ( T ) X , X2 X1 = = Є { x | x € X , f ( T ) x = 0 } ...
... converge for 0 ≤m < x ( 2 ) , and if lim , fn ( 20 ) 0 , then { f ( T ) } converges in the weak operator topology . Moreover , = x f ( T ) X { xx X , ƒ ( T ) x = 0 } . PROOF . Let 1 = ƒ ( T ) X , X2 X1 = = Є { x | x € X , f ( T ) x = 0 } ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ