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 = Corollary 14. Q.E.D. 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 * x x * y , x * ( x — y ) : 0 and x = Corollary 14. Q.E.D. y , by 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 293
... limit ¿ ( E ) = lim [ gn ( s ) u ( ds ) E exists for every E in 2o . By Lemma 8 the limit ( E ) exists for each E in 21 . Thus , by Theorems 6 and 7 , the sequence { gn } is weakly conver- gent in L ( S1 , E1 , u ) . Since L1 ( S1 , E1 ...
... limit ¿ ( E ) = lim [ gn ( s ) u ( ds ) E exists for every E in 2o . By Lemma 8 the limit ( E ) exists for each E in 21 . Thus , by Theorems 6 and 7 , the sequence { gn } is weakly conver- gent in L ( S1 , E1 , u ) . Since L1 ( S1 , E1 ...
Page 658
... limit ( iii ) T 1 lim T → ∞ + √ 1 ( 4 , ( 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 ...
... limit ( iii ) T 1 lim T → ∞ + √ 1 ( 4 , ( 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 ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ