Linear Operators: General theory |
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Page 143
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue measurable ...
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue measurable ...
Page 188
... measure has the property n = 1 n n μ ( PE ) - Π μ . ( Ε . ) , i = 1 i = 1 Ε , Ε Σ . , and it follows from the ... Lebesgue measure on the real line for i = 1 , ... , n . Then S = P1 S ; is n - dimensional Eucli- dean space , and м = MX ...
... measure has the property n = 1 n n μ ( PE ) - Π μ . ( Ε . ) , i = 1 i = 1 Ε , Ε Σ . , and it follows from the ... Lebesgue measure on the real line for i = 1 , ... , n . Then S = P1 S ; is n - dimensional Eucli- dean space , and м = MX ...
Page 223
... Lebesgue measure and that ƒ ' may vanish almost everywhere without f being a constant . 7 Let ( S , 2 , μ ) be the product of the regular measure spaces ( Sα , Zα , μɑ ) , α € A. If S is endowed with its product topology show that ( S ...
... Lebesgue measure and that ƒ ' may vanish almost everywhere without f being a constant . 7 Let ( S , 2 , μ ) be the product of the regular measure spaces ( Sα , Zα , μɑ ) , α € A. If S is endowed with its product topology show that ( S ...
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