Linear Operators: General theory |
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Page 68
... determined by { n } , and x ≤ lim inf x ,. 8 个 4 PROOF . If , in Theorem 6 , X , Y , T , are replaced by X * , Ø , în , it is seen that lim inf . Theorem 19 gives the inequality | x | ≤ ∞4U lim inf a ,, and Theorem 20 shows that sup ...
... determined by { n } , and x ≤ lim inf x ,. 8 个 4 PROOF . If , in Theorem 6 , X , Y , T , are replaced by X * , Ø , în , it is seen that lim inf . Theorem 19 gives the inequality | x | ≤ ∞4U lim inf a ,, and Theorem 20 shows that sup ...
Page 136
Nelson Dunford, Jacob T. Schwartz. 6 LEMMA . There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all ...
Nelson Dunford, Jacob T. Schwartz. 6 LEMMA . There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all ...
Page 455
... determined by GF . We can even assert that each continuous linear functional ƒ can be included in a denumerable self - determined set G of functionals . For , if ƒ is determined by the denumerable set G1 , let each functional in G1 be ...
... determined by GF . We can even assert that each continuous linear functional ƒ can be included in a denumerable self - determined set G of functionals . For , if ƒ is determined by the denumerable set G1 , let each functional in G1 be ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ