Linear Operators: General theory |
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Page 169
... zero in u - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L ( S , Σ , μ ) and that fn - f , converges to zero even if { f } is a general- ized sequence . 6 Let μ be bounded . Suppose that the field ...
... zero in u - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L ( S , Σ , μ ) and that fn - f , converges to zero even if { f } is a general- ized sequence . 6 Let μ be bounded . Suppose that the field ...
Page 204
... zero and since 0 ≤ fn ( $ x , ) ≤1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point so , in S , for which f ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
... zero and since 0 ≤ fn ( $ x , ) ≤1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point so , in S , for which f ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
Page 557
... zero polynomials S ,, i 2 , . . . , k such that S , ( T ) ; = 0 . If R S1 S2S , then R ( T ) , = 0 , and consequently R ( T ) x = 0 for all x X. Thus a non - zero polynomial R exists such that R ( T ) = • = 0 . Let R be factored as R ...
... zero polynomials S ,, i 2 , . . . , k such that S , ( T ) ; = 0 . If R S1 S2S , then R ( T ) , = 0 , and consequently R ( T ) x = 0 for all x X. Thus a non - zero polynomial R exists such that R ( T ) = • = 0 . Let R be factored as R ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ