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 , E1 , μ1 ) and ( S2 , 22 , 2 ) . For each E in Σ and s2 in S2 the set E ( $ 2 ) = { $ 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 , E1 , μ1 ) and ( S2 , 22 , 2 ) . For each E in Σ and s2 in S2 the set E ( $ 2 ) = { $ 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 ...
... 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 ...
Page 662
... Corollary II.3.13 that E'X is contained in the closure of ( I - 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 ( I - 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
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ