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 { x } of sca- lars such that n Σαπ lim | x − Σ αx , = 0 . 348 i = 1 : 00 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences y = { a } ...
... basis for X if to each xe X there corresponds a unique sequence { x } of sca- lars such that n Σαπ lim | x − Σ αx , = 0 . 348 i = 1 : 00 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences y = { a } ...
Page 254
... basis , the vector u has an expansion of the form ua Σβικα , ( u , v ) , so that u , is in the closed linear manifold de- termined by those v with ( u , v ) 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 ua Σβικα , ( u , v ) , so that u , is in the closed linear manifold de- termined by those v with ( u , v ) 0. Since such v are in V we have u € sp [ V ] and thus sp [ U ] C sp [ V ] ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ