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 ƒ on I is defined by the equation n v ( f , I ) = supΣf ...
... 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 ƒ on I is defined by the equation n v ( f , I ) = supΣf ...
Page 141
... interval I = ( a , b ) which may be finite or infinite . It is assumed that ( i ) f ( s ) = lim f ( s + e ) , = 04-3 se I ; i.e. , f is continuous on the right at every point in the open interval I. The closed interval Ī [ a , b ] is a ...
... interval I = ( a , b ) which may be finite or infinite . It is assumed that ( i ) f ( s ) = lim f ( s + e ) , = 04-3 se I ; i.e. , f is continuous on the right at every point in the open interval I. The closed interval Ī [ a , b ] is a ...
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 fag ( 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 fag ( s ) dh ( s ) exists . Let ƒ be a continuous increasing function on an open interval ( c , d ) ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ