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 * = 12 9 THEOREM . If 1 ≤ p ≤ ∞ and p - 1 + q - 1 = 1 , then the mapping [ a1 , ... , an ] determined by the equation x * x = n Σαβί x = { Pi } € ln , i = 1 is an isometric isomorphism of ( 1 " ) * onto ...
... gives the desired results : x * = 12 9 THEOREM . If 1 ≤ p ≤ ∞ and p - 1 + q - 1 = 1 , then the mapping [ a1 , ... , an ] determined by the equation x * x = n Σαβί x = { Pi } € ln , i = 1 is an isometric isomorphism of ( 1 " ) * onto ...
Page 287
... gives g * . On the other hand , Hölder's inequality ( III.3.2 ) gives | x * ≤ g ,. Thus | * | p The mapping * → 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 * . On the other hand , Hölder's inequality ( III.3.2 ) gives | x * ≤ g ,. Thus | * | p The mapping * → 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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ