Linear Operators: General theory |
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Page 26
... gives a third important and interesting way in which the concept of convergence can be generalized : 1 DEFINITION . A partially ordered set ( D , ≤ ) is said to be directed , if every finite subset of D has an upper bound . A map † : D ...
... gives a third important and interesting way in which the concept of convergence can be generalized : 1 DEFINITION . A partially ordered set ( D , ≤ ) is said to be directed , if every finite subset of D has an upper bound . A map † : D ...
Page 247
... gives the desired results : 9 THEOREM . If 1 ≤ p ≤ ∞ and p - 1 + q - 1 -1 x * → [ α1 , ... , an ] determined by the equation x * x n = Σαβίν 1 = 1 = x = { Pi } € ln , is an isometric isomorphism of ( 1 ) * onto la . 1 , then the ...
... gives the desired results : 9 THEOREM . If 1 ≤ p ≤ ∞ and p - 1 + q - 1 -1 x * → [ α1 , ... , an ] determined by the equation x * x n = Σαβίν 1 = 1 = x = { Pi } € ln , is an isometric isomorphism of ( 1 ) * onto la . 1 , then the ...
Page 287
... gives g , ≤ x * . On the other hand , Hölder's inequality ( III.3.2 ) gives * g . Thus * = g . The mapping a * → g is then a one - to - one isometric map of L * into L. It is evident from the Hölder inequality that any ge L , deter ...
... gives g , ≤ x * . On the other hand , Hölder's inequality ( III.3.2 ) gives * g . Thus * = g . The mapping a * → g is then a one - to - one isometric map of L * into L. It is evident from the Hölder inequality that any ge L , deter ...
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