Linear Operators: General theory |
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Page 186
... measure u of Lemma 1 is countably additive on D. Consequently , the restriction of u to the μ o - field generated by ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) = { 81 ...
... measure u of Lemma 1 is countably additive on D. Consequently , the restriction of u to the μ o - field generated by ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) = { 81 ...
Page 212
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
Page 725
... measure . ( Hint . Consider the map ƒ ( s ) → ZA ( S ) f ( ps ) for each Ae with μ ( 4 ) < ∞∞ . ) 37 Let ( S , E , μ ) be a positive measure space , and T a non - nega- tive linear transformation of L1 ( S , E , μ ) into itself ...
... measure . ( Hint . Consider the map ƒ ( s ) → ZA ( S ) f ( ps ) for each Ae with μ ( 4 ) < ∞∞ . ) 37 Let ( S , E , μ ) be a positive measure space , and T a non - nega- tive linear transformation of L1 ( S , E , μ ) into itself ...
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