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Page 288
... ( s ) = g ( s ) μ - almost everywhere on E , the limit g ( s ) lim , g ( s ) exists u - almost everywhere and vanishes ... Lp ( E ) is dense in L , by III.3.8 and the right and left sides of the last equation are both continuous in f , x * † = ...
... ( s ) = g ( s ) μ - almost everywhere on E , the limit g ( s ) lim , g ( s ) exists u - almost everywhere and vanishes ... Lp ( E ) is dense in L , by III.3.8 and the right and left sides of the last equation are both continuous in f , x * † = ...
Page 297
... ( S , Σ , μ ) with | x * | ≤ | 2 | . - If EĦ , i = 1 , 2 , . . . , n are disjoint sets in 2 with Σ12 ( E ) > 2ε , let ... Lp ( S , E , μ ) . Then for л == the characteristic function of { E1 , . . . , En } , we have n U_ƒ = ≥ { μ ( E ...
... ( S , Σ , μ ) with | x * | ≤ | 2 | . - If EĦ , i = 1 , 2 , . . . , n are disjoint sets in 2 with Σ12 ( E ) > 2ε , let ... Lp ( S , E , μ ) . Then for л == the characteristic function of { E1 , . . . , En } , we have n U_ƒ = ≥ { μ ( E ...
Page 302
... Lp ( S , E , μ ) is positive and write f≥ 0 if f ( s ) ≥ 0 for u - almost all se S. If f1 and f2 are two real or complex valued functions in L ( S , E , u ) such that f1 - f20 we write ff or fa≤ fi . This clearly is a partial ...
... Lp ( S , E , μ ) is positive and write f≥ 0 if f ( s ) ≥ 0 for u - almost all se S. If f1 and f2 are two real or complex valued functions in L ( S , E , u ) such that f1 - f20 we write ff or fa≤ fi . This clearly is a partial ...
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