Linear Operators: General theory |
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Page 239
... norm | x | = supα . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of all sequences x = { a } of scalars for which the norm is finite . x = ∞ က | x | = { Σ | αn | P } 1 / v n = 1 5. The space is the linear ...
... norm | x | = supα . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear space of all sequences x = { a } of scalars for which the norm is finite . x = ∞ က | x | = { Σ | αn | P } 1 / v n = 1 5. The space is the linear ...
Page 472
... norm in L ,, p > 1 is strongly differentiable at every point but the origin , and gave conditions for strong differentiability of the norm in the space L1 . He also showed that in the F - space of measurable functions on [ 0 , 1 ] , the ...
... norm in L ,, p > 1 is strongly differentiable at every point but the origin , and gave conditions for strong differentiability of the norm in the space L1 . He also showed that in the F - space of measurable functions on [ 0 , 1 ] , the ...
Page 532
... norm p / ( p − 1 ) , p > 1 , ( b ) ( Hilbert , Schur , Hardy , M. Riesz ) f ( y ) dy ( Tf ) ( x ) = Jo x + y is a map in L , ( 0 , ∞ ) of norm л ( sin л / р ) -1 , р > 1 , ( c ) ( Hardy , Littlewood , Polya ) ( T1 ) ( x ) = [ ° f ( y ) ...
... norm p / ( p − 1 ) , p > 1 , ( b ) ( Hilbert , Schur , Hardy , M. Riesz ) f ( y ) dy ( Tf ) ( x ) = Jo x + y is a map in L , ( 0 , ∞ ) of norm л ( sin л / р ) -1 , р > 1 , ( c ) ( Hardy , Littlewood , Polya ) ( T1 ) ( x ) = [ ° f ( y ) ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ