Linear Operators: General theory |
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Page 424
... closed . Hence TK = 0 , ye z A ( x , y ) 24,263 B ( z . r ) is also closed . Q.E.D. C 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space of the B - space X is compact in the X topology of X * . PROOF . By Definition ...
... closed . Hence TK = 0 , ye z A ( x , y ) 24,263 B ( z . r ) is also closed . Q.E.D. C 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space of the B - space X is compact in the X topology of X * . PROOF . By Definition ...
Page 429
... closed unit sphere of X is X - closed . PROOF . This follows from the preceding theorem and Corollary 2.14 . Q.E.D. 8 COROLLARY . If X is a B - space , a linear subspace YCX * is X - closed if and only if there exists an X - closed ...
... closed unit sphere of X is X - closed . PROOF . This follows from the preceding theorem and Corollary 2.14 . Q.E.D. 8 COROLLARY . If X is a B - space , a linear subspace YCX * is X - closed if and only if there exists an X - closed ...
Page 488
... closed range , then UX Y. ye Y and define PROOF . Let 0 Γ = { y * y * € * , y * y = 0 } . Then I is -closed in Y * . Suppose , for the moment , that U * T is X - closed and different from U ** . From Corollary V.3.12 it is seen that ...
... closed range , then UX Y. ye Y and define PROOF . Let 0 Γ = { y * y * € * , y * y = 0 } . Then I is -closed in Y * . Suppose , for the moment , that U * T is X - closed and different from U ** . From Corollary V.3.12 it is seen that ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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