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 * y , x * ( x - y ) = 0 and x Corollary 14. Q.E.D. x * x = y , by 27 LEMMA . A weakly convergent sequence { x } of points in a ...
... limit . PROOF . If x and y are both weak limits of a generalized sequence , then for each x * € X * , x * y , x * ( x - y ) = 0 and x Corollary 14. Q.E.D. x * x = y , by 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 658
... limit ( iii ) T 1 lim 1 ( þ1 ( 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 1 ( þ1 ( 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 , Σ , m ) has its norm | T | 1 1 ( Lemma 5.7 ) . Now let ƒ be a bounded μ ...
... 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 , Σ , m ) has its norm | T | 1 1 ( Lemma 5.7 ) . Now let ƒ be a bounded μ ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ