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Page 77
... converges . 48 Show that 00 ' 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 == { an } and quences of complex numbers , and let cm = b = ' m { b } be two ...
... converges . 48 Show that 00 ' 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 == { an } and quences of complex numbers , and let cm = b = ' m { b } be two ...
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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ