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 141
... interval I = ( a , b ) which may be finite or infinite . It is assumed that ( i ) f ( s ) = lim f ( s + | e | ) , sel : Ī i.e. , ƒ is continuous on the right at every point in the open interval I. The closed interval Ĩ = [ 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 | ) , sel : Ī i.e. , ƒ is continuous on the right at every point in the open interval I. The closed interval Ĩ = [ 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
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 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 disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ