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 S , 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 in L2 , Sn → I strongly for all c.o.n. ...
... Show that S , 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 in L2 , Sn → I strongly for all c.o.n. ...
Page 360
... Show that if ső E „ ( 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 ) | ≤M , < ∞ for each > 0 . 8 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 Sf for a given c.o.n. system is localized if and only if max E , ( x , y ) | ≤M , < ∞ for each > 0 . 8 22 Suppose that ( Snf ) ( x ) → f ( x ) uniformly for every f in ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ