Linear Operators: General theory |
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Page 169
... zero in μ - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L ( S , E , μ ) and that [ f - f , converges to zero even if { f } is a general- ized sequence . 6 Let μ be bounded . Suppose that the ...
... zero in μ - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L ( S , E , μ ) and that [ f - f , converges to zero even if { f } is a general- ized sequence . 6 Let μ be bounded . Suppose that the ...
Page 204
... zero and since 0 ≤ fn ( $ a , ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point so , in S , for which fn ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
... zero and since 0 ≤ fn ( $ a , ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point so , in S , for which fn ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
Page 595
... zero in the weak operator topology . Suppose that { fn ( T ) x } is weakly se- quentially compact for each x in X ... zero x in X2 , the sequence { fn ( T ) x } converges to a non - zero element of X2 . Since ƒ ( T ) commutes with R ( 2 ...
... zero in the weak operator topology . Suppose that { fn ( T ) x } is weakly se- quentially compact for each x in X ... zero x in X2 , the sequence { fn ( T ) x } converges to a non - zero element of X2 . Since ƒ ( T ) commutes with R ( 2 ...
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