Linear Operators: General theory |
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Page 239
... 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 norm 00 is finite . x = Ꮖ = ∞ n = 1 5. The space l is the linear space of all bounded ...
... 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 norm 00 is finite . x = Ꮖ = ∞ n = 1 5. The space l is the linear space of all bounded ...
Page 240
... norm { an } of n = sup | Σ | n i = 1 is finite . 11. The space cs is the linear space of all sequences x = { an } for which the series 1 , is convergent . The norm is n = 1 | x | = n sup | Σα .. n i = 1 12. Let S be an arbitrary set ...
... norm { an } of n = sup | Σ | n i = 1 is finite . 11. The space cs is the linear space of all sequences x = { an } for which the series 1 , is convergent . The norm is n = 1 | x | = n sup | Σα .. n i = 1 12. Let S be an arbitrary set ...
Page 532
... norm л ( sin л / р ) −1 , р > 1 , ( c ) ( Hardy , Littlewood , Polya ) ( Tf ) ( x ) = 8 10 f ( y ) dy max ( x , y ) is a map in L , ( 0 , ∞ ) of norm p2 ( p − 1 ) −1 , p > 1 . 22 by ( a ) Show that the mapping T : { a } → { b } of ...
... norm л ( sin л / р ) −1 , р > 1 , ( c ) ( Hardy , Littlewood , Polya ) ( Tf ) ( x ) = 8 10 f ( y ) dy max ( x , y ) is a map in L , ( 0 , ∞ ) of norm p2 ( p − 1 ) −1 , p > 1 . 22 by ( a ) Show that the mapping T : { a } → { b } of ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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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