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Page 188
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , 22 , μ2 ) . For each E in E and s2 in S , the set E ( 82 ) ...
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , 22 , μ2 ) . For each E in E and s2 in S , the set E ( 82 ) ...
Page 246
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , bn } is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , ... , n , defined by the equa- tions n x = Σb ...
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , bn } is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , ... , n , defined by the equa- tions n x = Σb ...
Page 662
... Corollary II.3.13 that E ' is contained in the closure of ( I - T ) X . Q.E.D. = 3 COROLLARY . If the sequence { 4 ( n ) } is bounded then it converges in the strong operator topology if and only if Tx / n converges to zero for x in a ...
... Corollary II.3.13 that E ' is contained in the closure of ( I - T ) X . Q.E.D. = 3 COROLLARY . If the sequence { 4 ( n ) } is bounded then it converges in the strong operator topology if and only if Tx / n converges to zero for x in a ...
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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