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 x = [ α1 , an ] of scalars .... 1 , ... , an with the norm | x | = supa . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of { a } of scalars for which the norm all ...
... linear space of all ordered n - tuples x = [ α1 , an ] of scalars .... 1 , ... , an with the norm | x | = supa . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of { a } of scalars for which the norm all ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ