Best Approximation in Normed Linear Spaces by Elements of by Prof. Ivan Singer (auth.)

By Prof. Ivan Singer (auth.)

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43). 6. 6. 11! = {g~, g~, ... 6. The method used in this proof (given, essentially, in [226], p. 10). 5. 42 Approximation by elements of arbitrary linear subspaces Chap. 1 Namely, let 11. (q)=max lx(q) - g;(q) I (i = 0, 1, ... 3. 54) i~O Then the sets y+, Y- are closed in Q and, by xEE'

F. Timan [250], p. 6 and in I. I. Ibragimov [92], p. 4. 6. APPLICATIONS IN THE SPACES C1(Q,v) AND ci(Q,v) Let Q be a compact space and v a positive Radon measure on Q. We shall denote by 0 1 (Q, v) the linear subspace of L 1 (Q, v) consisting of the equivalence classes of the (complex or real) continuous functions on Q, endowed with the usual vector operations and with the norm llxll = ~ lx(q) ld v(q). In theseQ quel we shall consider only the case when the carrier S( v) = Q; in this case each equivalence class of 0 1(Q,v) contains only one continuous function.

42) tJ. 43) max lx(t) - g0 (t) I for qEY+ teQ -max lx(t) - g0{t) I for qEY- (g 0 EM). 44) reQ Proof. Assume that we have MC~a(x). 4, there exists a real Radon measure tJ. 33). ); g0 , g~EM). 40) respectively. Then y+ and y- are well defined, disjoint and closed in Q. 6. 6 . e. 3). For this purpose we shall prove first some results on the maximal elements of continuous linear functionals on Cn(Q) which are also of interest for other applications. Let feCn(Q)*. 11 llxll is called a maximal element of f; in general, f need not have any maximal element or it may have several linearly independent maximal elements.

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