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 = y , by Corollary 14. Q.E.D. x * x = 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 = y , by Corollary 14. Q.E.D. x * x = 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 126
... limit inferior and the limit superior of { E } by the equations lim inf En - n 00 ∞ Em n = 1 m = n ' n 00 00 lim sup EU Em⋅ n En n = 1 m = n If lim inf , E = lim sup , En , { E } is said to be convergent , and we write the common ...
... limit inferior and the limit superior of { E } by the equations lim inf En - n 00 ∞ Em n = 1 m = n ' n 00 00 lim sup EU Em⋅ n En n = 1 m = n If lim inf , E = lim sup , En , { E } is said to be convergent , and we write the common ...
Page 293
... limit 2 ( E ) = = lim fn ( s ) μ ( ds ) 8 个 4 E n exists for every E in 2o . By Lemma 8 the limit λ ( E ) exists for each E in 21 . Thus , by Theorems 6 and 7 , the sequence { g } is weakly conver- gent in L1 ( S1 , E1 , μ ) . Since L1 ...
... limit 2 ( E ) = = lim fn ( s ) μ ( ds ) 8 个 4 E n exists for every E in 2o . By Lemma 8 the limit λ ( E ) exists for each E in 21 . Thus , by Theorems 6 and 7 , the sequence { g } is weakly conver- gent in L1 ( S1 , E1 , μ ) . Since L1 ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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