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 a is orthogonal to A. Thus E is the orthogonal ...
... 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 a is orthogonal to A. Thus E is the orthogonal ...
Page 578
... independent . Suppose that a .... n - 1 are linearly independent , but that a1 = 11 + ... .. Am ... + -1-1 . Then n 0 = ( T - λI ) xn = α1 ( 21 - λn ) x1 + ... + an - 1 ( 2n - 1 - λn ) xn − 1 and , since -λn 0 for in , we have α ; = 0 ...
... independent . Suppose that a .... n - 1 are linearly independent , but that a1 = 11 + ... .. Am ... + -1-1 . Then n 0 = ( T - λI ) xn = α1 ( 21 - λn ) x1 + ... + an - 1 ( 2n - 1 - λn ) xn − 1 and , since -λn 0 for in , we have α ; = 0 ...
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