Linear Operators: General theory |
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Page 136
... unique . PROOF . Theorem 4 and Lemma 5 show that the outer measure μ is one non - negative countably additive extension of μ to the o - field Zo determined by 2. Suppose that u1 is another such extension . If μ is o - finite on to prove ...
... unique . PROOF . Theorem 4 and Lemma 5 show that the outer measure μ is one non - negative countably additive extension of μ to the o - field Zo determined by 2. Suppose that u1 is another such extension . If μ is o - finite on to prove ...
Page 202
... unique . This uniqueness argument suggests how we may prove the existence of μ . For each finite set л in A there is , by Lemma 1 , a unique additive set function μ , defined on 2 , which has the property μ2 ( PE ) = II μa ( Ex ) , Π ...
... unique . This uniqueness argument suggests how we may prove the existence of μ . For each finite set л in A there is , by Lemma 1 , a unique additive set function μ , defined on 2 , which has the property μ2 ( PE ) = II μa ( Ex ) , Π ...
Page 516
... unique up to a positive constant factor if and only if n - 1 Σf ( p ( s ) ) converges uniformly to a constant for ... unique regular measure μ defined for the Borel subsets of S such that μ ( þ − 1 ( E ) ) = μ ( E ) for each Borel ...
... unique up to a positive constant factor if and only if n - 1 Σf ( p ( s ) ) converges uniformly to a constant for ... unique regular measure μ defined for the Borel subsets of S such that μ ( þ − 1 ( E ) ) = μ ( E ) for each Borel ...
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