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 N μ ( PE ) = Π μ . ( Ε . ) , i = 1 i = 1 Ε , Ε Σ . , and it follows from the uniqueness ... Lebesgue measure on the real line for i = 1 , ... , n . Then S = P1 S , is n - dimensional Eucli- dean space , and μ ...
... measure has the property n N μ ( PE ) = Π μ . ( Ε . ) , i = 1 i = 1 Ε , Ε Σ . , and it follows from the uniqueness ... Lebesgue measure on the real line for i = 1 , ... , n . Then S = P1 S , is n - dimensional Eucli- dean space , and μ ...
Page 223
... Lebesgue measure and that f ' may vanish almost everywhere without f being a constant . 7 Let ( S , 2 , μ ) be the product of the regular measure spaces ( Sα , Za , Ma ) , α € A. If S is endowed with its product topology show that ( S ...
... Lebesgue measure and that f ' may vanish almost everywhere without f being a constant . 7 Let ( S , 2 , μ ) be the product of the regular measure spaces ( Sα , Za , Ma ) , α € A. If S is endowed with its product topology show that ( S ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ