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 Y for which U * y * = 0 implies y * y = 0 . PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then , since U , has a dense range , ** € X * is in the ...
... range , then the range of U is closed and consists of those vectors y in Y for which U * y * = 0 implies y * y = 0 . PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then , since U , has a dense range , ** € X * is in the ...
Page 514
... range . 20 A projection has finite dimensional range if and only if it is compact . = 21 A linear mapping E such that E2 E is a projection ( i.e. , is bounded ) if and only if the ranges of E and I - E are closed . Let E be a projection ...
... range . 20 A projection has finite dimensional range if and only if it is compact . = 21 A linear mapping E such that E2 E is a projection ( i.e. , is bounded ) if and only if the ranges of E and I - E are closed . Let E be a projection ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ