Linear Operators: General theory |
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Page 131
... continuous with respect to u or simply u - continuous , if lim 2 ( E ) 0 . - v ( u , E ) → 0 The function is said to be μ - singular if there is a set Ee such that v ( u , Eo ) = 2 ( E ) = λ ( EEO ) , ΕΕΣ . It is clear that the only ...
... continuous with respect to u or simply u - continuous , if lim 2 ( E ) 0 . - v ( u , E ) → 0 The function is said to be μ - singular if there is a set Ee such that v ( u , Eo ) = 2 ( E ) = λ ( EEO ) , ΕΕΣ . It is clear that the only ...
Page 315
... continuous if and only if λ2 is μ2 - continuous . Ալ PROOF . Clearly if 22 is 2 - continuous , λ , is μ - continuous . To prove the converse we recall ( cf. the remarks following Definition III.4.12 ) that it is sufficient to show ...
... continuous if and only if λ2 is μ2 - continuous . Ալ PROOF . Clearly if 22 is 2 - continuous , λ , is μ - continuous . To prove the converse we recall ( cf. the remarks following Definition III.4.12 ) that it is sufficient to show ...
Page 513
... continuous with either the uniform or weak operator topology . By considering the sequence { 4 } defined in Exercise 11 , show that this mapping is not continuous in the strong operator topology . 13 If U : Y * → X * is a linear ...
... continuous with either the uniform or weak operator topology . By considering the sequence { 4 } defined in Exercise 11 , show that this mapping is not continuous in the strong operator topology . 13 If U : Y * → X * is a linear ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ