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 f ...
... 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 f ...
Page 247
... gives the desired results : x * 9 THEOREM . If 1 ≤ p ≤ ∞o and p - 1 + q - 1 an ] determined by the equation n - 00 1 , then the mapping x * x = Σαιβιο Ꮖ - { fi } € ln , i = 1 is an isometric isomorphism of ( ln ) * onto la . PROOF ...
... gives the desired results : x * 9 THEOREM . If 1 ≤ p ≤ ∞o and p - 1 + q - 1 an ] determined by the equation n - 00 1 , then the mapping x * x = Σαιβιο Ꮖ - { fi } € ln , i = 1 is an isometric isomorphism of ( ln ) * onto la . PROOF ...
Page 287
... gives ga * . On the other hand , Hölder's inequality ( III.3.2 ) gives x * ≤g ,. Thus a * = g│ ,. Р 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 g e L ...
... gives ga * . On the other hand , Hölder's inequality ( III.3.2 ) gives x * ≤g ,. Thus a * = g│ ,. Р 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 g e L ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ