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Page 49
... field , it is to be under- stood that the statement applies to both the real and the complex cases . 1 DEFINITION . A group G is said to be a topological group if : ( i ) G is a Hausdorff space ; ( ii ) The mapping ( x , y ) → xy - 1 ...
... field , it is to be under- stood that the statement applies to both the real and the complex cases . 1 DEFINITION . A group G is said to be a topological group if : ( i ) G is a Hausdorff space ; ( ii ) The mapping ( x , y ) → xy - 1 ...
Page 137
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that | μ ( C ) | < ɛ for every C in Σ with CCG - F . For a complex or extended real ...
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that | μ ( C ) | < ɛ for every C in Σ with CCG - F . For a complex or extended real ...
Page 168
... field , and clearly the smallest field containing . An elementary induction shows that C , n = 1 , 2 , . . . , is countable and thus Σ is a countable field . Q.E.D. ท 5 LEMMA . Let ( S , E , μ ) be a positive measure space and G a ...
... field , and clearly the smallest field containing . An elementary induction shows that C , n = 1 , 2 , . . . , is countable and thus Σ is a countable field . Q.E.D. ท 5 LEMMA . Let ( S , E , μ ) be a positive measure space and G a ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ