Linear Operators: General theory |
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Page 36
... 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 space is said to be finite dimen- sional . In a ...
... 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 space is said to be finite dimen- sional . In a ...
Page 71
... basis for X if to each xe X there corresponds a unique sequence { a } of sca- lars such that € n lim | x − Σ x , x = 0 . 24∞ i = 1 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences y = { x } ...
... basis for X if to each xe X there corresponds a unique sequence { a } of sca- lars such that € n lim | x − Σ x , x = 0 . 24∞ i = 1 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences y = { x } ...
Page 254
... basis , the vector u , has an expansion of the form u = ( u , v ) vg , so that u is in the closed linear manifold de- termined by those vg with ( u , v ) UB α B 0. Since such v are in V we have u 。€ sp [ V ] and thus sp [ U ] C sp [ V ] ...
... basis , the vector u , has an expansion of the form u = ( u , v ) vg , so that u is in the closed linear manifold de- termined by those vg with ( u , v ) UB α B 0. Since such v are in V we have u 。€ sp [ V ] and thus sp [ U ] C sp [ V ] ...
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