Linear Operators: General theory |
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Page 424
... closed . Hence тK = ( x , yex A ( x , y ) Ca € z € 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 тK = ( x , yex A ( x , y ) Ca € z € 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 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. PROOF . Let 0ye Y and define Γ r = { y * y * € Y * , y * y = 0 } . Then I is -closed in 9 * . Suppose , for the moment , that U * T is X - closed and different from U * Y * . From Corollary V.3.12 it is seen ...
... closed range , then UX = Y. PROOF . Let 0ye Y and define Γ r = { y * y * € Y * , y * y = 0 } . Then I is -closed in 9 * . Suppose , for the moment , that U * T is X - closed and different from U * Y * . From Corollary V.3.12 it is seen ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ