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. = 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 * x x * y , x * ( x - y ) = 0 and x = y , by Corollary 14. Q.E.D. = 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 127
... limit 1 En . A monotone sequence is one which is either non - decreasing or non - increasing . We note that the intersection , union , limit inferior , and limit superior of every sequence of measurable sets is measurable . 4 LEMMA ...
... limit 1 En . A monotone sequence is one which is either non - decreasing or non - increasing . We note that the intersection , union , limit inferior , and limit superior of every sequence of measurable sets is measurable . 4 LEMMA ...
Page 658
... limit ( iii ) lim 1 - T iS 1 ( s f ( þt ( x ) ) dt , 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 ...
... limit ( iii ) lim 1 - T iS 1 ( s f ( þt ( x ) ) dt , 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 ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ