Linear Operators: General theory |
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Page 228
... derivatives of fn of arbitrarily large order converge to the corresponding partial derivatives of f uniformly in V. From this theorem , we can easily derive the useful fact that if f is analytic in the Cartesian product U1X ... XU2 of a ...
... derivatives of fn of arbitrarily large order converge to the corresponding partial derivatives of f uniformly in V. From this theorem , we can easily derive the useful fact that if f is analytic in the Cartesian product U1X ... XU2 of a ...
Page 384
... derivatives including those of order n converge uniformly to the corresponding derivatives of f . ) Of the many extensions of the Weierstrass theorem for [ 0 , 1 ] we mention the striking theorem of Müntz [ 1 ] which asserts that the ...
... derivatives including those of order n converge uniformly to the corresponding derivatives of f . ) Of the many extensions of the Weierstrass theorem for [ 0 , 1 ] we mention the striking theorem of Müntz [ 1 ] which asserts that the ...
Page 655
... derivatives and with second derivatives f " lying in L ,, and which is defined by Aff " , ƒ e D ( A ) . 16 Let X = L „ ( 0 , ∞ ) , 1 ≤ p < ∞ . Let T ( t ) ( x , s ) = x ( t + s ) , eX . Show T ( t ) is a strongly VIII.3.13 655 EXERCISES.
... derivatives and with second derivatives f " lying in L ,, and which is defined by Aff " , ƒ e D ( A ) . 16 Let X = L „ ( 0 , ∞ ) , 1 ≤ p < ∞ . Let T ( t ) ( x , s ) = x ( t + s ) , eX . Show T ( t ) is a strongly VIII.3.13 655 EXERCISES.
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