Linear Operators: General theory |
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Page 36
... independent set . The cardinality of a Hamel basis is a number independent of the Hamel basis ; it is called the dimension of the linear space . This independence is readily proved if there is a finite Hamel basis , in which case the ...
... independent set . The cardinality of a Hamel basis is a number independent of the Hamel basis ; it is called the dimension of the linear space . This independence is readily proved if there is a finite Hamel basis , in which case the ...
Page 252
... independent of the order of its terms . Now it is clear that E is a linear operator with Ex = a for x in A. Thus Ex = x for x in the closed linear manifold A1 determined by A. Also Ex = 0 if x is orthogonal to A. Thus E is the ...
... independent of the order of its terms . Now it is clear that E is a linear operator with Ex = a for x in A. Thus Ex = x for x in the closed linear manifold A1 determined by A. Also Ex = 0 if x is orthogonal to A. Thus E is the ...
Page 578
... independent . Suppose that a , ... , n - 1 are linearly independent , but that 1 = 12 + ... + an - 1-1 . Then - 0 = ( T - 2,1 ) x1 = α1 ( 21 - λn ) x1 + ... + αn - 1 ( 2n - 1 - λn ) Xn - 1 · and , since 2 - λn ‡ 0 for i ‡ n , we have ...
... independent . Suppose that a , ... , n - 1 are linearly independent , but that 1 = 12 + ... + an - 1-1 . Then - 0 = ( T - 2,1 ) x1 = α1 ( 21 - λn ) x1 + ... + αn - 1 ( 2n - 1 - λn ) Xn - 1 · and , since 2 - λn ‡ 0 for i ‡ n , we have ...
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