Linear Operators: General theory |
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Page 305
... e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , -1821 be decompositions of f1 and f2 into g1i + Σg2 ; is a decomposition of f1 + f2 , and let f1 = -161i f2 = fi 12 positive functions . Then and so To ( fi + f2 ) ≤ To ( f1 ) + To ( f2 ) . n ...
... e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , -1821 be decompositions of f1 and f2 into g1i + Σg2 ; is a decomposition of f1 + f2 , and let f1 = -161i f2 = fi 12 positive functions . Then and so To ( fi + f2 ) ≤ To ( f1 ) + To ( f2 ) . n ...
Page 320
... E into a finite number of disjoint sets in Σ . A number of elementary properties of the semi - variation are listed in the next lemma . 4 LEMMA . Let μ be a vector valued measure . Then ( a ) || μ || ( E ) ... Fi , ... , k j = • . " " k , we ...
... E into a finite number of disjoint sets in Σ . A number of elementary properties of the semi - variation are listed in the next lemma . 4 LEMMA . Let μ be a vector valued measure . Then ( a ) || μ || ( E ) ... Fi , ... , k j = • . " " k , we ...
Page 627
... fi ( t ) = e - wtf ( t ) , so that f1e L1 ( 0 , ∞ ) and sof1 ( t ) e - 1t dt R ( 2 ) ≥0 . We make the change of variable u = e - t . Then t = — log u , and fo et f ( t ) dt = fu3g ( u ) du , where g ( u ) = u ( — logu ) .is in L1 [ 0 ...
... fi ( t ) = e - wtf ( t ) , so that f1e L1 ( 0 , ∞ ) and sof1 ( t ) e - 1t dt R ( 2 ) ≥0 . We make the change of variable u = e - t . Then t = — log u , and fo et f ( t ) dt = fu3g ( u ) du , where g ( u ) = u ( — logu ) .is in L1 [ 0 ...
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