Linear Operators: General theory |
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Page 20
... assumed that no A , contains a non - void open set . Thus A1 X , and A ' is open , and contains a sphere S1 S ( P1 , 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non ...
... assumed that no A , contains a non - void open set . Thus A1 X , and A ' is open , and contains a sphere S1 S ( P1 , 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non ...
Page 177
Nelson Dunford, Jacob T. Schwartz. may be assumed that 2 is real valued . A real valued set function can be represented as the ... assumed to be positive and measurable we can put F = { ss e F , f ( s ) ≤ n } . Then F = F and v ( λ , F ...
Nelson Dunford, Jacob T. Schwartz. may be assumed that 2 is real valued . A real valued set function can be represented as the ... assumed to be positive and measurable we can put F = { ss e F , f ( s ) ≤ n } . Then F = F and v ( λ , F ...
Page 278
... assumed that x * } = x * f . Then there is a point s in S such that x * f = f ( s ) , fe C ( S ) . PROOF . By the preceding lemma a * is a point in the space S1 de- fined in the proof of Theorem 18. By Theorem 22 , S is homeomorphic to ...
... assumed that x * } = x * f . Then there is a point s in S such that x * f = f ( s ) , fe C ( S ) . PROOF . By the preceding lemma a * is a point in the space S1 de- fined in the proof of Theorem 18. By Theorem 22 , S is homeomorphic to ...
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