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 | x | = sup . 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 ...
... linear space of all ordered n - tuples [ α1 , ... , an ] of scalars α1 , ... , an with the norm | x | = sup . 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 ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ