Mass- und Integrationstheorie by Bauer H.

By Bauer H.

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Let Xl, X2 E dom f. Then (Xl, f(Xl)) E epi f and (X2, f(X2)) E epi f. 2 (8), find Ql(Xl, f(xI) + Q2(X2, f(X2)) E epi f. e. tl 2:: f(xd and t2 2:: f(X2) (if dom f = 0 then f(x) = += (x E X) and epi f = 0). 2 = 1. 6. DEFINITION. A mapping p : X ~ JR.. is a sublinear functional provided that epi p is a cone. 7. If dom p i= then the following statements are equivalent: (1) p is a sublinear functional; (2) p is a convex function that is positively homogeneous: p(Qx) = Qp(x) for all 0. + and Xl, X2 E X, then P(QlXl + Q2 X2) :::; QlP(Xt} + Q2P(X2); (4) p is a positively homogeneous functional that is subadditive: P(XI + X2) :::; p(xt} + p(X2) (Xl, X2 EX).

9. Every equicontinuous set consists of uniformly continuous mappings. Every finite set of uniformly continuous mappings is equicontinuous. 3. 1. ¥:= Xl x X 2 • Given x:= (Xl, X2) and y:= (Yl, Y2), put Tben d is a metric on :Z". Moreover, for every X:= (Xl, X2) in :Z" tbe presentation bolds: TX(X) = fil{Ul X U2 : Ul E TXl(Xl), U2 E TX 2 (X2)}. 2. DEFINITION. The topology TX is called the product of TX l and TX 2 or the product topology of Xl x X 2. 3. DEFINITION. A function f : X -+ lR" is lower semicontinuous if its epigraph epi f is closed in the product topology of X x JR.

9. Are there any analogs for a Hamel basis in general modules? 10. When does a sum of projections present a projection itself? 11. Let T be an endomorphism of some vector space which satisfies the conditions Tn-l I- 0 and Tn 0 for a natural n. Prove that the operators TO, T, ... ,Tn-l are linearly independent. 12. Describe the structure of a linear operator defined on the direct sum of spaces and acting into the product of spaces. 13. Find conditions for unique solvability of the following equations in operators ~A B and A~= B (here the operator ~ is unknown).

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