Linear Operators: General theory |
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Page 4
... lim sup A and lim inf A are usually denoted by lim inf an lim sup an n1∞ " ∞4U respectively . If a and b are extended real numbers , then the symbol ( a , b ) denotes the open interval defined by { x | a < x < b } ; the symbol [ a , b ] ...
... lim sup A and lim inf A are usually denoted by lim inf an lim sup an n1∞ " ∞4U respectively . If a and b are extended real numbers , then the symbol ( a , b ) denotes the open interval defined by { x | a < x < b } ; the symbol [ a , b ] ...
Page 73
... LIM s , = x * ( s ) , show that ( a ) LIM s = LIM 8n + 15 N1∞ n - ∞ ∞48 ( b ) LIM ( as , + ẞtn ) = a LIM sn + ẞ LIM tn ; 21∞ ∞4U n1∞ ( c ) LIM s , 0 if [ s ] is a non - negative sequence ; ( d ) lim inf s , ≤ LIM s , ≤ lim sup s ...
... LIM s , = x * ( s ) , show that ( a ) LIM s = LIM 8n + 15 N1∞ n - ∞ ∞48 ( b ) LIM ( as , + ẞtn ) = a LIM sn + ẞ LIM tn ; 21∞ ∞4U n1∞ ( c ) LIM s , 0 if [ s ] is a non - negative sequence ; ( d ) lim inf s , ≤ LIM s , ≤ lim sup s ...
Page 299
... lim * + _ ^ \ f ( y ) dy = 0 uniformly for ƒ € K. fe PROOF . Suppose K is ... Sup- posing this , it follows that all the functions g1 , EN vanish outside ... lim fg , ( x + y ) —g , ( y ) dy = 0 for each function g1 ; and hence 04 - x • + ...
... lim * + _ ^ \ f ( y ) dy = 0 uniformly for ƒ € K. fe PROOF . Suppose K is ... Sup- posing this , it follows that all the functions g1 , EN vanish outside ... lim fg , ( x + y ) —g , ( y ) dy = 0 for each function g1 ; and hence 04 - x • + ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ