Linear Operators: General theory |
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Page 424
... closed . Hence tK = α , xex B ( x , x ) is also closed . Q.E.D. ^ x , veX A ( x , y ) 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 = α , xex B ( x , x ) is also closed . Q.E.D. ^ x , veX A ( x , y ) 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 488
... closed range , then UX = Y. PROOF . Let 0 y € Y and define r = { y * y * € Y * , y * y = 0 } . Then I is -closed in Y * . J ) Suppose , for the moment , that U * I 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 r = { y * y * € Y * , y * y = 0 } . Then I is -closed in Y * . J ) Suppose , for the moment , that U * I 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 U1X Hence , U has a closed range . Q.E.D. - UX - 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range . PROOF . Let y = lim ...
... closed . It follows from the previous lemma that U1X Hence , U has a closed range . Q.E.D. - UX - 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range . PROOF . Let y = lim ...
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