Linear Operators: General theory |
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Page 279
... shows that │Hƒ | ≤ | ƒ | and proves the H is continuous . To see that his continuous , let N be a neighborhood of the point so h ( to ) . By Theorems I.5.2 and 1.5.9 there is a continuous function f on S with f ( so ) = 1 and f ( s ) ...
... shows that │Hƒ | ≤ | ƒ | and proves the H is continuous . To see that his continuous , let N be a neighborhood of the point so h ( to ) . By Theorems I.5.2 and 1.5.9 there is a continuous function f on S with f ( so ) = 1 and f ( s ) ...
Page 285
... show that the sum f + g of two almost periodic functions is almost periodic , also shows that the product fg is almost periodic . Thus the present theorem is a corollary of Theorem 6.18 . Q.E.D. 8. The Spaces L ( S , E , μ ) The spaces ...
... show that the sum f + g of two almost periodic functions is almost periodic , also shows that the product fg is almost periodic . Thus the present theorem is a corollary of Theorem 6.18 . Q.E.D. 8. The Spaces L ( S , E , μ ) The spaces ...
Page 574
... shows that ( 21 - T ) ' x = ( 1 - T ) ' e ( T ) x = 0 , and proves that the index n ≤ v . Q.E.D. T 19 THEOREM . Let f be in F ( T ) , and let τ be a spectral set of f ( T ) . Then o ( T ) f - 1 ( t ) is a spectral set of T , and = E ...
... shows that ( 21 - T ) ' x = ( 1 - T ) ' e ( T ) x = 0 , and proves that the index n ≤ v . Q.E.D. T 19 THEOREM . Let f be in F ( T ) , and let τ be a spectral set of f ( T ) . Then o ( T ) f - 1 ( t ) is a spectral set of T , and = E ...
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