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 x for x in A. Thus Ex = x for x in the closed linear manifold A , determined by A. Also Ex = 0 if a 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 x for x in A. Thus Ex = x for x in the closed linear manifold A , determined by A. Also Ex = 0 if a is orthogonal to A. Thus E is the ...
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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ