Linear Operators: General theory |
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Page 169
... Show that there need not exist any 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 certain ...
... Show that there need not exist any 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 certain ...
Page 358
... Show that the range of S , lies in 1 ≤ p ≤∞ , C ( ∞ ) . k = n 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 ...
... Show that the range of S , lies in 1 ≤ p ≤∞ , C ( ∞ ) . k = n 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 ...
Page 360
... Show that if fő En ( x , z ) dz ≤ M , then the convergence of Sf for a given c.o.n. system is localized if and only if max | E , ( x , y ) 1x - 12 € M. < ∞ for each ɛ > 0 . 8 22 Suppose that ( Sf ) ( x ) f ( x ) uniformly for every f ...
... Show that if fő En ( x , z ) dz ≤ M , then the convergence of Sf for a given c.o.n. system is localized if and only if max | E , ( x , y ) 1x - 12 € M. < ∞ for each ɛ > 0 . 8 22 Suppose that ( Sf ) ( x ) f ( x ) uniformly for every f ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ