Linear Operators: General theory |
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Page 186
... measure μ of Lemma 1 is countably additive on P. Consequently , the ... space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the ... positive measure spaces ( S1 , E1 , μ1 ) and ( S2 , Z2 , μg ) . For each E ...
... measure μ of Lemma 1 is countably additive on P. Consequently , the ... space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the ... positive measure spaces ( S1 , E1 , μ1 ) and ( S2 , Z2 , μg ) . For each E ...
Page 302
... positive measure space . Then the real partially ordered space L , ( S , Σ , μ ) , 1 ≤ p < ∞ , is a complete lattice . PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fago for some go ...
... positive measure space . Then the real partially ordered space L , ( S , Σ , μ ) , 1 ≤ p < ∞ , is a complete lattice . PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fago for some go ...
Page 304
... positive measure space and let 1≤ p ≤∞o . Suppose f1 + f2 Σ - 18 , where f , gx are positive elements in L , ( S ... space and let 1 ≤ p ≤ ∞ and 1 ≤ q < ∞o . Then the partially ordered space of linear mappings from the real space ...
... positive measure space and let 1≤ p ≤∞o . Suppose f1 + f2 Σ - 18 , where f , gx are positive elements in L , ( S ... space and let 1 ≤ p ≤ ∞ and 1 ≤ q < ∞o . Then the partially ordered space of linear mappings from the real space ...
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