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 [ fn - 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 [ fn - 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 ( 8 ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point in S for which fr ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero . Thus ...
... zero and since 0 ≤ fn ( 8 ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point in S for which fr ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero . Thus ...
Page 231
... zero , it is seen that f ( z ) M everywhere in the strip . If f is analytic in a connected open set U of the complex plane and not identically zero , U contains no point which is the limit of zeros of f . This fact can be concluded from ...
... zero , it is seen that f ( z ) M everywhere in the strip . If f is analytic in a connected open set U of the complex plane and not identically zero , U contains no point which is the limit of zeros of f . This fact can be concluded from ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ