Linear Operators: General theory |
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Page 239
... norm = sup [ x ] . 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 ∞ │x = { Σ │α , P } 1 / v n = 1 is finite . 5. The space 00 x = { x } of scalars ...
... norm = sup [ x ] . 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 ∞ │x = { Σ │α , P } 1 / v n = 1 is finite . 5. The space 00 x = { x } of scalars ...
Page 240
... norm { n } of is finite . n | x | : = sup | Σ | n i = 1 11. The space cs is the linear space of all sequences x = for which the series 1 is convergent . The norm is n | x | = sup | Σα .. n { an } i = 1 12. Let S be an arbitrary set ...
... norm { n } of is finite . n | x | : = sup | Σ | n i = 1 11. The space cs is the linear space of all sequences x = for which the series 1 is convergent . The norm is n | x | = sup | Σα .. n { an } i = 1 12. Let S be an arbitrary set ...
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 ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ