Linear Operators: General theory |
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Page 488
... closed range , then UX Y. PROOF . Let 0 Γ ye Y and define r = { y * y * € Y * , y * y = 0 } . Then I is -closed in * . Suppose , for the moment , that U * T is X - closed and different . from U ** . From Corollary V.3.12 it is seen that ...
... closed range , then UX Y. PROOF . Let 0 Γ ye Y and define r = { y * y * € Y * , y * y = 0 } . Then I is -closed in * . Suppose , for the moment , that U * T is X - closed and different . from U ** . From Corollary V.3.12 it is seen that ...
Page 489
... closed . It follows from the previous lemma that U1X Hence , U has a closed range . Q.E.D. UX -- 1 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 -- 1 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 ...
Page 513
... closed if there exists a constant K such that for any y in the range there exists a solution of y Tx such that x ... closed subspace of a B - space and N is a finite di- mensional subspace , then Y N is a closed subspace . If Y N is a ...
... closed if there exists a constant K such that for any y in the range there exists a solution of y Tx such that x ... closed subspace of a B - space and N is a finite di- mensional subspace , then Y N is a closed subspace . If Y N is a ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ