Linear Operators: General theory |
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Page 58
... continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the corresponding topologies contains the other , the two topologies are equal ...
... continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the corresponding topologies contains the other , the two topologies are equal ...
Page 131
... continuous with respect to μ or simply u - continuous , if lim 2 ( E ) = v ( μ , E ) →→ 0 0 . The function ¿ is said to be u - singular if there is a set Ee such that v ( μ , Eo ) = 0 ; 2 ( E ) = 2 ( EE 。) , ΕΕΣ . It is clear that ...
... continuous with respect to μ or simply u - continuous , if lim 2 ( E ) = v ( μ , E ) →→ 0 0 . The function ¿ is said to be u - singular if there is a set Ee such that v ( μ , Eo ) = 0 ; 2 ( E ) = 2 ( EE 。) , ΕΕΣ . It is clear that ...
Page 315
... continuous if and only if 22 is μ - continuous . PROOF . Clearly if λ is μ - continuous , 1 is μ1 - continuous . To prove the converse we recall ( cf. the remarks following Definition III.4.12 ) that it is sufficient to show that if v ...
... continuous if and only if 22 is μ - continuous . PROOF . Clearly if λ is μ - continuous , 1 is μ1 - continuous . To prove the converse we recall ( cf. the remarks following Definition III.4.12 ) that it is sufficient to show that if v ...
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