Linear Operators: General theory |
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Page 424
... closed . Hence TK = ( x , yex A ( x , y ) a € , xe X B ( α , 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 = ( x , yex A ( x , y ) a € , xe X B ( α , 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 488
... closed range , then UX = Y. PROOF . Let 0 y € Y and define Γ T = { y * y * € Y * , y * y = 0 } . Then I is -closed in 9 * . Suppose , for the moment , that U * is X - closed and different from U ** . From Corollary V.3.12 it is seen ...
... closed range , then UX = Y. PROOF . Let 0 y € Y and define Γ T = { y * y * € Y * , y * y = 0 } . Then I is -closed in 9 * . Suppose , for the moment , that U * is X - closed and different from U ** . From Corollary V.3.12 it is seen ...
Page 489
... closed . It follows from the previous lemma that UX Hence , U has a closed range . Q.E.D. = U * y * for some y * e Y * . If 2 * is U ** . Hence , the range of U is = * UX 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed ...
... closed . It follows from the previous lemma that UX Hence , U has a closed range . Q.E.D. = U * y * for some y * e Y * . If 2 * is U ** . Hence , the range of U is = * UX 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ