By Prof. Ivan Singer (auth.)

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**Example text**

29). Define Y:. 39) and Y;;. 40). 38). 37). In fact, if p. were not non-decreasing on there would exist a set AcY:. J. J. 29). 38). J. 38). JnYt. (q) teQ + 36 Approximation by elements of arbitrary linear subspaces +~ S(IL)nr;o = ( max lx(t) )Q IEQ Chap. e. 29). 29) which completes the proof. Consider now the problem of simultaneous characterization of a set M cG of elements of best approximation. 34). 5, it follows that if MC~a(x), then there exist a (common) real Radon measure tJ. 38) for all g0 EM.

W. Cheney and A. A. Goldstein ( [35 ], theorem 1) : Let L be a real topological linear space, Q a compact subset of L and x a continuous real-valued function on Q. 59) In fact, let E = On(Q}, let u: L*-+E be defined by u(z*) = = z* IQ (z*EL*) and let G = u(L*) cE, g0 = u(z~}EG. 40) respectively. 1-o on A. J. o (q) = ~Q [ u(z*) ](q)d[L(q) = 0 (z*eL*), and thus*) 0 belongs to the closed convex hull of the set A.. o on A. J. 0 (q) =O(z*eL*). 37), which completes the proof. For similar results on the characterization and existence of z~eL* see B.

13. 9 has been given in [230], theorem 5, and the equivalence r~6° in the paper [234], §1. 111) presents interest only as a necessary condition for g 0 e~a(x), since in the sufficiency parts it may be replaced by llf0 11 =1. 10. Let E be a normed linear space, G a linear subspace of E, xeE"- G and g0 e~a(x). u. c. 105). 66 Approximation by elements of arbitrary linear subspaces Proof. For every yeE, let us denote by bounded function on*) S(8m,x,u0 ) defined by y(f) = y the Chap. ), under the mapping y-+y).