Linear Operators: General theory |
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Page 140
... intervals . An interval is finite if both its end points are finite ; otherwise it is an infinite interval . If ƒ is a complex function on an interval I , the total variation of f on I is defined by the equation v ( f , I ) = n sup Σ f ...
... intervals . An interval is finite if both its end points are finite ; otherwise it is an infinite interval . If ƒ is a complex function on an interval I , the total variation of f on I is defined by the equation v ( f , I ) = n sup Σ f ...
Page 223
... interval ( a , b ) and continuous on the right . Let g be a function defined on ( a , b ) such that the Lebesgue - Stieltjes integral I Sag ( s ) dh ( s ) exists . Let ƒ be a continuous increasing function on an open interval ( c , d ) ...
... interval ( a , b ) and continuous on the right . Let g be a function defined on ( a , b ) such that the Lebesgue - Stieltjes integral I Sag ( s ) dh ( s ) exists . Let ƒ be a continuous increasing function on an open interval ( c , d ) ...
Page 283
... interval x ≤1 . By the same rea- soning { f } contains a subsequence { f } convergent uniformly on the interval 2. In this way successive subsequences are chosen so that the limit lim ; fn , ¿ ( x ) exists uniformly on the interval | x ...
... interval x ≤1 . By the same rea- soning { f } contains a subsequence { f } convergent uniformly on the interval 2. In this way successive subsequences are chosen so that the limit lim ; fn , ¿ ( x ) exists uniformly on the interval | 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