Linear Operators: General theory |
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Page 151
... follows immediately from Theorem 15 , as conditions ( i ) and ( ii ) are satisfied by { f } . Q.E.D. 17 COROLLARY . Let ( S , E , μ ) be a positive measure space , and let { f } be a monotone increasing sequence of non - negative real ...
... follows immediately from Theorem 15 , as conditions ( i ) and ( ii ) are satisfied by { f } . Q.E.D. 17 COROLLARY . Let ( S , E , μ ) be a positive measure space , and let { f } be a monotone increasing sequence of non - negative real ...
Page 189
... follows immediately from Corollary 6.17 . Q.E.D. In proving the crucial Theorem 9 below , we will have use for the following easy consequence of Corollary 7 . 8 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite ...
... follows immediately from Corollary 6.17 . Q.E.D. In proving the crucial Theorem 9 below , we will have use for the following easy consequence of Corollary 7 . 8 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite ...
Page 634
... follows immediately that p ( E ) is a Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have 1 + 00 • + ∞ + ∞ [ +2 [ + % ( E ) ( s , t ) ds dt = [ + ] % ( st ) dsdt 00 ...
... follows immediately that p ( E ) is a Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have 1 + 00 • + ∞ + ∞ [ +2 [ + % ( E ) ( s , t ) ds dt = [ + ] % ( st ) dsdt 00 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ