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 . 13 COROLLARY . ( a ) If ( S , Σ , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , E , μ ) is a finite measure space , an ...
... everywhere . 13 COROLLARY . ( a ) If ( S , Σ , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , E , μ ) is a finite measure space , an ...
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 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