Linear Operators: General theory |
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Page 171
... Suppose that every continuous function is u - measurable . Show that * contains every Borel set . 24 Suppose that S is a compact topological space , with the property that for every covering of S by a finite number of open sets G1 ...
... Suppose that every continuous function is u - measurable . Show that * contains every Borel set . 24 Suppose that S is a compact topological space , with the property that for every covering of S by a finite number of open sets G1 ...
Page 225
... suppose that B consists of a finite collection of disjoint closed rectifiable Jordan curves ; that is , we suppose that B can be decomposed into the union B = B1UBU ... UB of dis- joint closed sets B , in such a way that each B , is a ...
... suppose that B consists of a finite collection of disjoint closed rectifiable Jordan curves ; that is , we suppose that B can be decomposed into the union B = B1UBU ... UB of dis- joint closed sets B , in such a way that each B , is a ...
Page 360
... Suppose that 0≤x≤27 ( i ) ( Snf ) ( x ) → f ( x ) uniformly in æ for ƒ in AC . ( ii ) The convergence of Sf is localized . Show that for any ƒ in CBV , ( Sf ) ( x ) f ( x ) uniformly in a . - > 24 Suppose ( i ) , ( ii ) of the last ...
... Suppose that 0≤x≤27 ( i ) ( Snf ) ( x ) → f ( x ) uniformly in æ for ƒ in AC . ( ii ) The convergence of Sf is localized . Show that for any ƒ in CBV , ( Sf ) ( x ) f ( x ) uniformly in a . - > 24 Suppose ( i ) , ( ii ) of the last ...
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