Linear Operators: General theory |
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Page 169
... Show that there need not exist any set Ee containing A such that v ( μ , E ) = 0 . 3 Suppose that S is the interval ... Show that a real valued function fis u - measurable if and only if it is continuous at every point not lying in a ...
... Show that there need not exist any set Ee containing A such that v ( μ , E ) = 0 . 3 Suppose that S is the interval ... Show that a real valued function fis u - measurable if and only if it is continuous at every point not lying in a ...
Page 358
... Show that the range of S1 lies in C ( ∞ ) . n • 3 Show that S1 → I strongly in any one of the spaces C ( k ) , k < ∞ , AC , L „ , 1 ≤ p < ∞ if and only if | S „ | ≤K , where | S „ | de- notes the operator norm of S , in the space ...
... Show that the range of S1 lies in C ( ∞ ) . n • 3 Show that S1 → I strongly in any one of the spaces C ( k ) , k < ∞ , AC , L „ , 1 ≤ p < ∞ if and only if | S „ | ≤K , where | S „ | de- notes the operator norm of S , in the space ...
Page 360
... Show that if fő En ( x , z ) dz | ≤ M , then the convergence of Snf for a given c.o.n. system is localized if and only if max | E , ( x , y ) | ≤M , < ∞ for each ɛ > 0 . 8 | xv | ≥ € 22 Suppose that ( S „ f ) ( x ) → f ( x ) ...
... Show that if fő En ( x , z ) dz | ≤ M , then the convergence of Snf for a given c.o.n. system is localized if and only if max | E , ( x , y ) | ≤M , < ∞ for each ɛ > 0 . 8 | xv | ≥ € 22 Suppose that ( S „ f ) ( x ) → f ( x ) ...
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