Linear Operators: General theory |
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Page 46
... equal to the cardinality of B and if the cardinality of B is at most equal to the cardinality of 4 , then these two sets have the same cardinality . ) 3 If X is a vector space over Ø , 46 I.14.1 I. PRELIMINARY CONCEPTS Exercises.
... equal to the cardinality of B and if the cardinality of B is at most equal to the cardinality of 4 , then these two sets have the same cardinality . ) 3 If X is a vector space over Ø , 46 I.14.1 I. PRELIMINARY CONCEPTS Exercises.
Page 111
... equal . PROOF . Using Lemma 15 , - Sefndu - Set du ≤sf ; ( 8 ) − fm ( 8 ) | v ( μ , ds ) → 0 , E E m i = 1 , 2 . Thus , the desired limits exist uniformly with respect to E in Σ and it only remains to show that they are equal . In ...
... equal . PROOF . Using Lemma 15 , - Sefndu - Set du ≤sf ; ( 8 ) − fm ( 8 ) | v ( μ , ds ) → 0 , E E m i = 1 , 2 . Thus , the desired limits exist uniformly with respect to E in Σ and it only remains to show that they are equal . In ...
Page 336
... equal μ - almost everywhere if and only if 0 ( A ( ƒ1 ) ) and 0 ( A ( ƒ1⁄2 ) ) are equal . Now the o - finite space ( T , 21 , 0 ) can contain at most a countable number of disjoint measur- able sets of non - zero 0 - measure , and so ...
... equal μ - almost everywhere if and only if 0 ( A ( ƒ1 ) ) and 0 ( A ( ƒ1⁄2 ) ) are equal . Now the o - finite space ( T , 21 , 0 ) can contain at most a countable number of disjoint measur- able sets of non - zero 0 - measure , and so ...
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