Linear Operators: General theory |
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Page 169
... Show that there need not exist any set Ee containing A such that v ( u , 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 ( u , 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 ( ' ( ∞ ) . n 3 Show that S → I strongly in any one of the spaces C ( * ) , k < ∞ , AC , L , 1 ≤ p < ∞ if and only if S , ≤ K , where S , de- notes the operator norm of S , in the space . Show ...
... Show that the range of S1 lies in ( ' ( ∞ ) . n 3 Show that S → I strongly in any one of the spaces C ( * ) , k < ∞ , AC , L , 1 ≤ p < ∞ if and only if S , ≤ K , where S , de- notes the operator norm of S , in the space . Show ...
Page 360
... Show that if | ső E „ ( x , z ) dz | ≤M , then the convergence of Snf for a given c.o.n. system is localized if and only if max En ( x , y ) ! M. < ∞ for each ɛ > 0 . 22 Suppose that ( Snf ) ( x ) → f ( x ) uniformly for every f in ...
... Show that if | ső E „ ( x , z ) dz | ≤M , then the convergence of Snf for a given c.o.n. system is localized if and only if max En ( x , y ) ! M. < ∞ for each ɛ > 0 . 22 Suppose that ( Snf ) ( x ) → f ( x ) uniformly for every f in ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ