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 489
... range of U is also closed . It follows from the previous lemma that UX 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 ...
... range of U is also closed . It follows from the previous lemma that UX 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 ...
Page 513
... range of U is closed . ( ii ) The range of U is closed if there exists a constant K such that for any y in the range there exists a solution of y x ≤ Ky . - Tx such that ( iii ) U is one - to - one if the range of U * is dense in X ...
... range of U is closed . ( ii ) The range of U is closed if there exists a constant K such that for any y in the range there exists a solution of y x ≤ Ky . - Tx such that ( iii ) U is one - to - one if the range of U * is dense in X ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ