Linear Operators: General theory |
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Page 68
... limit . Ꮖ PROOF . If x and y are both weak limits of a generalized sequence , then for each x * X * , x * x = 0 and x = y , by Corollary 14. Q.E.D. = x * y , x * ( x - y ) = 27 LEMMA . A weakly convergent sequence { x } of points in a ...
... limit . Ꮖ PROOF . If x and y are both weak limits of a generalized sequence , then for each x * X * , x * x = 0 and x = y , by Corollary 14. Q.E.D. = x * y , x * ( x - y ) = 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 126
... limit inferior and the limit superior of { E } by the equations 00 00 x 00 lim inf E U Em lim sup En nu Em · n = 1 m = n n If lim inf , En - lim n = 1 m = n supn En , { En } is said to be convergent , and we write the common value of the ...
... limit inferior and the limit superior of { E } by the equations 00 00 x 00 lim inf E U Em lim sup En nu Em · n = 1 m = n n If lim inf , En - lim n = 1 m = n supn En , { En } is said to be convergent , and we write the common value of the ...
Page 127
... limit " n = 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 " n = 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 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ