Linear Operators: General theory |
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Page 131
... continuous with respect to u or simply μ - continuous , if lim 2 ( E ) = 0 . v ( μ , E ) → 0 The function is said to be μ - singular if there is a set E e Σ such that v ( μ , E。) = 0 ; λ ( E ) = 2 ( EE 。) , ΕΕΣ . It is clear that the ...
... continuous with respect to u or simply μ - continuous , if lim 2 ( E ) = 0 . v ( μ , E ) → 0 The function is said to be μ - singular if there is a set E e Σ such that v ( μ , E。) = 0 ; λ ( E ) = 2 ( EE 。) , ΕΕΣ . It is clear that the ...
Page 315
... continuous if and only if λ2 is μ2 - continuous . that PROOF . Clearly if λ is μ2 - continuous , λ , is μ1 - 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 . that PROOF . Clearly if λ is μ2 - continuous , λ , is μ1 - 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 ** is a linear mapping ...
... 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 ** is a linear mapping ...
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