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 - ax ∞48 Σαπ = 0 . i = 1 i = 1 x ; ¤ ; 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences ...
... basis for X if to each xe X there corresponds a unique sequence { a ; } of sca- lars such that € n lim x - ax ∞48 Σαπ = 0 . i = 1 i = 1 x ; ¤ ; 8 Let { x } be a basis in the F - space X , and let Y be the vector space of all sequences ...
Page 254
... basis , the vector u has an expansion of the ( u , v ) , so that u , is in the closed linear manifold de- α form u = 0. Since such are in V we have Similarly sp [ V ] ≤ sp [ U ] . It is thus termined by those ϋβ with ( u , v ) ue sp ...
... basis , the vector u has an expansion of the ( u , v ) , so that u , is in the closed linear manifold de- α form u = 0. Since such are in V we have Similarly sp [ V ] ≤ sp [ U ] . It is thus termined by those ϋβ with ( u , v ) ue sp ...
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