Linear Operators: General theory |
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Page 68
... limit . PROOF . If x and y are both weak limits of a generalized sequence , then for each x * X * , x * x x * y , x * ( x - y ) 0 and x = y , by Corollary 14. Q.E.D. = - LEMMA . A weakly convergent sequence { x } of points in a normed ...
... limit . PROOF . If x and y are both weak limits of a generalized sequence , then for each x * X * , x * x x * y , x * ( x - y ) 0 and x = y , by Corollary 14. Q.E.D. = - LEMMA . A weakly convergent sequence { x } of points in a normed ...
Page 658
... limit . ( iii ) T 1 lim f ( p ( x ) ) dt , T which , in physical theories , is assumed to exist . Thus , the mathematician is led to the problem of determining whether or not the limit ( iii ) exists . The next four sections are devoted ...
... limit . ( iii ) T 1 lim f ( p ( x ) ) dt , T which , in physical theories , is assumed to exist . Thus , the mathematician is led to the problem of determining whether or not the limit ( iii ) exists . The next four sections are devoted ...
Page 683
... limit m = lim ca ( 2 , u ) . By Corollary 5.2 , m ( p - 1e ) = m ( e ) , so that the map T : ƒ ( · ) → f ( p ( ) ) as an operator in the space L1 ( S , E , m ) has its norm T1 = 1 ( Lemma 5.7 ) . Now let ƒ be a bounded u - measurable ...
... limit m = lim ca ( 2 , u ) . By Corollary 5.2 , m ( p - 1e ) = m ( e ) , so that the map T : ƒ ( · ) → f ( p ( ) ) as an operator in the space L1 ( S , E , m ) has its norm T1 = 1 ( Lemma 5.7 ) . Now let ƒ be a bounded u - measurable ...
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