Linear Operators: General theory |
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Page 186
... measure μ of Lemma 1 is countably additive on Ø . Consequently , the restriction of u to the o - field generated by ... positive measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and 82 in S2 the set E ( 82 ) = { 81 ] ...
... measure μ of Lemma 1 is countably additive on Ø . Consequently , the restriction of u to the o - field generated by ... positive measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and 82 in S2 the set E ( 82 ) = { 81 ] ...
Page 212
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each FF has positive u - measure and there is a positive ...
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each FF has positive u - measure and there is a positive ...
Page 304
... positive measure space and let 1 ≤ p ≤oo . Suppose f1 + f2 1g , where fi , gx are positive elements in L „ ( S , Σ , μ ) . Then there are positive elements h ̧1⁄2 in L ( S , Σ , μ ) , j = 1 , 2 , k 1 , ... , n , such that n jk f1 = Σ ...
... positive measure space and let 1 ≤ p ≤oo . Suppose f1 + f2 1g , where fi , gx are positive elements in L „ ( S , Σ , μ ) . Then there are positive elements h ̧1⁄2 in L ( S , Σ , μ ) , j = 1 , 2 , k 1 , ... , n , such that n jk f1 = Σ ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ