Linear Operators: General theory |
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Page 487
... Range It was observed in Lemma 2.8 that the closure of the range of an operator U € B ( X , Y ) consists of those vectors y such that y * UX = 0 implies y * y 0. Or , in other words , Є = UX = { y U * y * - 0 implies y * y = 0 } . The ...
... Range It was observed in Lemma 2.8 that the closure of the range of an operator U € B ( X , Y ) consists of those vectors y such that y * UX = 0 implies y * y 0. Or , in other words , Є = UX = { y U * y * - 0 implies y * y = 0 } . The ...
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 . Y - PROOF . Consider the map U1 from X to 3 = U ( X ) , defined by 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 . Y - PROOF . Consider the map U1 from X to 3 = U ( X ) , defined by U1 ( x ) = U ( x ) . Then , since U1 has a dense range ...
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 Tx such that │x ≤ K│y . = ( 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 Tx such that │x ≤ K│y . = ( iii ) U is one - to - one if the range of U * is dense in X ...
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