Linear Operators: General theory |
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Page 364
... Exercises 35 and 49. ) 51 Show that if Tm is the operator of Exercise 45 , ( Tmf ) ( x ) → f ( x ) at every point of the Lebesgue set of a function fe L. Show that the same holds for the operator T , of Exercise 47. ( Hint . Use ...
... Exercises 35 and 49. ) 51 Show that if Tm is the operator of Exercise 45 , ( Tmf ) ( x ) → f ( x ) at every point of the Lebesgue set of a function fe L. Show that the same holds for the operator T , of Exercise 47. ( Hint . Use ...
Page 365
... Exercise 53 maps H , in a linear one - one manner onto the closed subspace of L , consisting of those F all of whose negative Fourier coefficients vanish . 55 Using the notations of Exercises 53 and 54 , show that if fe H , and if ( U ...
... Exercise 53 maps H , in a linear one - one manner onto the closed subspace of L , consisting of those F all of whose negative Fourier coefficients vanish . 55 Using the notations of Exercises 53 and 54 , show that if fe H , and if ( U ...
Page 371
... Exercise 85 to apply to zeros on the boundary of the unit disc . ) 88 89 Show that Exercise 87 is valid even if p - 1 . Every function f in H1 can be written as a product gh , where g and h are in H2 . ( Hint : Use Exercise 88. ) 90 ...
... Exercise 85 to apply to zeros on the boundary of the unit disc . ) 88 89 Show that Exercise 87 is valid even if p - 1 . Every function f in H1 can be written as a product gh , where g and h are in H2 . ( Hint : Use Exercise 88. ) 90 ...
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