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 f is a complex function on an interval I , the total variation of ƒ 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 f is a complex function on an interval I , the total variation of ƒ on I is defined by the equation v ( f , I ) = n sup Σ f ...
Page 141
... interval ( a , b ) which may be finite or infinite . It is assumed that ( i ) f ( s ) = lim f ( s + ε ' ) , 04-3 sel ; 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 ( a , b ) which may be finite or infinite . It is assumed that ( i ) f ( s ) = lim f ( s + ε ' ) , 04-3 sel ; 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 283
... interval | x | ≤ 1. By the same rea- soning { fi } contains a subsequence { f2 ,; } 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 ...
... interval | x | ≤ 1. By the same rea- soning { fi } contains a subsequence { f2 ,; } 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 ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ