Linear Operators: General theory |
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Page 151
... Consequently , lim sup \ fn - fmp≤ lim sup { { £ , \ † n ( 8 ) — † m ( 8 ) | P v ( μ , ds ) } 1 ” + 2ɛ1 » m , n∞ m ... consequently fe L , and fn - fp0 . Q.E.D. 16 COROLLARY . ( Lebesgue dominated convergence theorem ) Let 1 ≤ p ...
... Consequently , lim sup \ fn - fmp≤ lim sup { { £ , \ † n ( 8 ) — † m ( 8 ) | P v ( μ , ds ) } 1 ” + 2ɛ1 » m , n∞ m ... consequently fe L , and fn - fp0 . Q.E.D. 16 COROLLARY . ( Lebesgue dominated convergence theorem ) Let 1 ≤ p ...
Page 451
... Consequently , letting Zn , i , j = · { x \ x € X , j { { ( x + yn / j ) −2t ( x ) + t ( x− ¥ n / j ) } < we have Z1 = U Zn no Zn , i , ; . If we put Zn , i = UZn , i , j , then = 021 Zn Zn , i is open in X , and Zn = 1Zn , i We wish ...
... Consequently , letting Zn , i , j = · { x \ x € X , j { { ( x + yn / j ) −2t ( x ) + t ( x− ¥ n / j ) } < we have Z1 = U Zn no Zn , i , ; . If we put Zn , i = UZn , i , j , then = 021 Zn Zn , i is open in X , and Zn = 1Zn , i We wish ...
Page 627
... Consequently , by the Weierstrass approximation theorem , and Corollary III.10.6 , = 0 = f ' h ( u ) g ( u ) du = [ ' h ( u ) G ( du ) , he C ( 0 , 1 ) , where G ( E ) SEg ( u ) du . Since Lebesgue measure is regular so is G and thus ...
... Consequently , by the Weierstrass approximation theorem , and Corollary III.10.6 , = 0 = f ' h ( u ) g ( u ) du = [ ' h ( u ) G ( du ) , he C ( 0 , 1 ) , where G ( E ) SEg ( u ) du . Since Lebesgue measure is regular so is G and thus ...
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