Linear Operators: General theory |
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Page 143
... Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure determined by a func- tion f of bounded variation on a finite or infinite interval is the Le- besgue extension of the Borel ...
... Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure determined by a func- tion f of bounded variation on a finite or infinite interval is the Le- besgue extension of the Borel ...
Page 634
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have • + ∞ √ √ 100 % . ( E ) ( s , t ) ds dt -1 80 - ∞ + J + ∞ [ + € [ + ZE ( s − t ) dsdt -∞ 00 - -∞ ∞- ∞ + J c ...
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have • + ∞ √ √ 100 % . ( E ) ( s , t ) ds dt -1 80 - ∞ + J + ∞ [ + € [ + ZE ( s − t ) dsdt -∞ 00 - -∞ ∞- ∞ + J c ...
Page 838
... Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) in a partially ordered set , I.2.3 ( 4 ) in the ( extended ) real ...
... Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) in a partially ordered set , I.2.3 ( 4 ) in the ( extended ) real ...
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