Linear Operators: General theory |
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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 ( S1 , 1 , 1 ) and ( S , 2 , 2 ) . For each E in 2 and s in S2 the set E ( 82 ) = { $ 1 ...
... 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 ( S1 , 1 , 1 ) and ( S , 2 , 2 ) . For each E in 2 and s in S2 the set E ( 82 ) = { $ 1 ...
Page 246
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , b } is a Hamel basis for the normed linear space X then the functionals b ‡ , i = 1 , . . . , n , defined by the equa- tions n x = Σ ...
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , b } is a Hamel basis for the normed linear space X then the functionals b ‡ , i = 1 , . . . , n , defined by the equa- tions n x = Σ ...
Page 662
... Corollary II.3.13 that E'X is contained in the closure of ( 1 - T ) X . Q.E.D. 3 COROLLARY . If the sequence { A ( n ) } is bounded then it converges in the strong operator topology if and only if Trx / n converges to zero for x in a ...
... Corollary II.3.13 that E'X is contained in the closure of ( 1 - T ) X . Q.E.D. 3 COROLLARY . If the sequence { A ( n ) } is bounded then it converges in the strong operator topology if and only if Trx / n converges to zero for x in a ...
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