Linear Operators: General theory |
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Page 4
... sup , + ∞ inf 4. If A is an infinite set of real numbers , then the symbol lim sup 4 denotes the infimum of all numbers b with the property that only a finite set of numbers in A exceed b ; the de- finition of the symbol lim inf A is ...
... sup , + ∞ inf 4. If A is an infinite set of real numbers , then the symbol lim sup 4 denotes the infimum of all numbers b with the property that only a finite set of numbers in A exceed b ; the de- finition of the symbol lim inf A is ...
Page 73
... lim inf sn ≤ LIM 8 个袋 Sn lim sup s if [ sn ] is a real sequence ; 8 个袋( e ) If sn is a convergent sequence , then - LIM Sn lim Sn · ∞4U N1∞ 23 Show that there exists a method of assigning a " generalized limit " LIM to every ...
... lim inf sn ≤ LIM 8 个袋 Sn lim sup s if [ sn ] is a real sequence ; 8 个袋( e ) If sn is a convergent sequence , then - LIM Sn lim Sn · ∞4U N1∞ 23 Show that there exists a method of assigning a " generalized limit " LIM to every ...
Page 299
... lim [ + f ( y ) \ o dy = 0 uniformly for f € K. A · 00 PROOF . Suppose K is ... Sup- posing this , it follows that all the functions g1 , . . EN vanish ... lim ( + \ x ( x + y ) —x ( y ) | Pdy = 0 81 if % is the characteristic function of ...
... lim [ + f ( y ) \ o dy = 0 uniformly for f € K. A · 00 PROOF . Suppose K is ... Sup- posing this , it follows that all the functions g1 , . . EN vanish ... lim ( + \ x ( x + y ) —x ( y ) | Pdy = 0 81 if % is the characteristic function of ...
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