Linear Operators: General theory |
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Page 487
... range UX is closed , in which case the range U ** is also closed . Dually , if U ** is closed , so is UX . These results are contained in the next two theorems . Addi- tional information along these lines is to be found in the set of ...
... range UX is closed , in which case the range U ** is also closed . Dually , if U ** is closed , so is UX . These results are contained in the next two theorems . Addi- tional information along these lines is to be found in the set of ...
Page 488
... range , then the range of U is closed and consists of those vectors y in for which U * y * = 0 implies y * y = 0 . 1 = U ( X ) , defined by PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then , since U1 has a dense range ...
... range , then the range of U is closed and consists of those vectors y in for which U * y * = 0 implies y * y = 0 . 1 = U ( X ) , defined by PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then , since U1 has a dense range ...
Page 489
... range of U * is also closed . It follows from the previous lemma that U1X = UX = 3 . Hence , U has a closed range . Q.E.D. 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range ...
... range of U * is also closed . It follows from the previous lemma that U1X = UX = 3 . Hence , U has a closed range . Q.E.D. 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ