Linear Operators: General theory |
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Page 89
... linear spaces which are not complete . In such cases , the following theorem is often ... linear space satisfying properties ( i ) and ( ii ) of Definition 1.10 . Then X is isomorphic and isometric with a dense linear subspace of an F - space ...
... linear spaces which are not complete . In such cases , the following theorem is often ... linear space satisfying properties ( i ) and ( ii ) of Definition 1.10 . Then X is isomorphic and isometric with a dense linear subspace of an F - space ...
Page 91
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen ...
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen ...
Page 239
... linear space of all ordered n - tuples [ α1 , .. an of scalars α1 , . , an with the norm = supα . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of all sequences x = { n } of scalars for which the norm is finite ...
... linear space of all ordered n - tuples [ α1 , .. an of scalars α1 , . , an with the norm = supα . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of all sequences x = { n } of scalars for which the norm is finite ...
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