Linear Operators: General theory |
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Page 100
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
Page 150
... everywhere . ( b ) If ( S , E , μ ) is a finite measure space , an almost everywhere convergent sequence of measurable functions is convergent in μ - measure . 14 COROLLARY . If ( S , Z , μ ) is a measure space , and { fn } is a ...
... everywhere . ( b ) If ( S , E , μ ) is a finite measure space , an almost everywhere convergent sequence of measurable functions is convergent in μ - measure . 14 COROLLARY . If ( S , Z , μ ) is a measure space , and { fn } is a ...
Page 722
... everywhere . 1 22 Let T be a transformation in L , T≥0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is real , fe L1 , and supos A ( n ) f ≥0 almost everywhere , and info≤n A ( n ) f≤0 almost everywhere , then ƒ = 0 . 23 ...
... everywhere . 1 22 Let T be a transformation in L , T≥0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is real , fe L1 , and supos A ( n ) f ≥0 almost everywhere , and info≤n A ( n ) f≤0 almost everywhere , then ƒ = 0 . 23 ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ