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 a ex there corresponds a unique sequence { a } of sca- lars such that n lim x - ax 12∞ Σαπ € 0 . i = 1 8 Let { x } be a basis in the F - space X , and let 9 be the vector space of all sequences y { a } for which ...
... basis for X if to each a ex there corresponds a unique sequence { a } of sca- lars such that n lim x - ax 12∞ Σαπ € 0 . i = 1 8 Let { x } be a basis in the F - space X , and let 9 be the vector space of all sequences y { a } for which ...
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 vs 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 u ( u , v ) vg , so that u , is in the closed linear manifold de- termined by those vs with ( u , v ) 0. Since such v are in V we have u € sp [ V ] and thus sp [ U ] C sp [ V ] ...
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