Linear Operators: General theory |
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Page 145
... uniformly on S— E. The sequence { f } converges μ - 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 converges uniformly to f on S - E . It is clear that u - uniform convergence ...
... uniformly on S— E. The sequence { f } converges μ - 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 converges uniformly to f on S - E . It is clear that u - uniform convergence ...
Page 314
... uniformly for λą € VK . It is then clear that u V - 1 ( μ2 ) is a non - nega- tive element of ba ( S , Z ) such that lim ( E ) = 0 uniformly for € K . λ μ ( E ) -0 To prove the converse , suppose there exists a non - negative ue ba ( S ...
... uniformly for λą € VK . It is then clear that u V - 1 ( μ2 ) is a non - nega- tive element of ba ( S , Z ) such that lim ( E ) = 0 uniformly for € K . λ μ ( E ) -0 To prove the converse , suppose there exists a non - negative ue ba ( S ...
Page 360
... uniformly for every f in AC . Show that there exists a finite constant K such that for ƒ in CBV , | ( Snf ) ( x ) | ≤K ( v ( f , [ 0 , 27 ] ) + sup | f ( x ) ) , 0 ≤ x ≤ 2ñ . 23 Suppose that ( i ) ( Sf ) ( x ) → f ( x ) uniformly in ...
... uniformly for every f in AC . Show that there exists a finite constant K such that for ƒ in CBV , | ( Snf ) ( x ) | ≤K ( v ( f , [ 0 , 27 ] ) + sup | f ( x ) ) , 0 ≤ x ≤ 2ñ . 23 Suppose that ( i ) ( Sf ) ( x ) → f ( x ) uniformly in ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 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