Linear Operators: General theory |
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Page 4
... sup , + ∞ info . If A is an infinite set of real numbers , then the symbol lim sup A 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 , + ∞ info . If A is an infinite set of real numbers , then the symbol lim sup A 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 sn = = 1 , x * ( e ) = 1 , x * ( s ) , show that ∞48 ( a ) LIM sn = LIM Sn + 15 N - ∞ n∞ ( b ) LIM ( as , + ẞtn ) = α LIM sn + ẞ LIM t2 ; 348 Na 21∞ ( c ) LIM sn≥ 0 if [ s ] is a non - negative sequence ; 7x ( d ) lim inf s , ≤ ...
... LIM sn = = 1 , x * ( e ) = 1 , x * ( s ) , show that ∞48 ( a ) LIM sn = LIM Sn + 15 N - ∞ n∞ ( b ) LIM ( as , + ẞtn ) = α LIM sn + ẞ LIM t2 ; 348 Na 21∞ ( c ) LIM sn≥ 0 if [ s ] is a non - negative sequence ; 7x ( d ) lim inf s , ≤ ...
Page 299
... lim [ * + [ ~ * † ( y ) Þ dy = 0 uniformly for ƒ € K. 844 A PROOF . Suppose ... Sup- posing this , it follows that all the functions g1 , . . . , gy vanish ... lim ( x ( x + y ) —z ( y ) \ ” dy = 0 81 if x is the characteristic function ...
... lim [ * + [ ~ * † ( y ) Þ dy = 0 uniformly for ƒ € K. 844 A PROOF . Suppose ... Sup- posing this , it follows that all the functions g1 , . . . , gy vanish ... lim ( x ( x + y ) —z ( y ) \ ” dy = 0 81 if x is the characteristic function ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ