Linear Operators: General theory |
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Page 424
... closed . Hence TK = 0 , yeX A ( x , y ) Oɑ € , æ € X B ( x , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space X * of the B - space X is compact in the X topology of X * . PROOF . By ...
... closed . Hence TK = 0 , yeX A ( x , y ) Oɑ € , æ € X B ( x , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space X * of the B - space X is compact in the X topology of X * . PROOF . By ...
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 YC X * 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 YC X * is X - closed if and only if there exists an X - closed ...
Page 488
... closed range , then UX - Y. PROOF . Let 0ye Y and define Γ = { y * y * € Y * , y * y = 0 } . Then I is -closed in Y * . Ux = € = = Y as desired . = 0 , Suppose , for the moment , that U * T is X - closed and different from U ** . From ...
... closed range , then UX - Y. PROOF . Let 0ye Y and define Γ = { y * y * € Y * , y * y = 0 } . Then I is -closed in Y * . Ux = € = = Y as desired . = 0 , Suppose , for the moment , that U * T is X - closed and different from U ** . From ...
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