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Extra resources for A First Course in Infinitesimal Calculus
C Q such that we have d[J. (q) = d! [1. i x(q)-g 0 (q) max] x(t) -g0 (t)! IE Q Consequently, for any pair g0 , g~ E 1W we have hence whence*') (q E S([L)). 33). 4. 4. "\:Ve mention that another simultaneous characterization of a set McG(cC(Q)) of elements of best approximation, less convenient for applications, has been given by Z. S. Romanova (, theorem 1). 30). **) See N. Bourbaki , Chap. III, p. 73, corollary of proposition 10, and p. 70, proposition 2. 4. APPLICATIONS IN THE SPACES CR(Q) For a compact space Q, we shall denote by 0 R( Q) the space of all continuous real-valued functions on Q, endowed with the usual vector operations and with the norm Jixll =max lx(q)l.
Proof ( , p. 335). Let g0 E §fa(x) and let g be an arbitrary element of G. 113). 113). Then for every g E G we have Jlx-goll = Ret(x- g0 ) :::;; Rej"(x- g):::;; I f'(x- g) I :::;; :::;; llf"llllx- g II = II x- g)), whence g0E§£ 0 (x), which completes the proof. 13. 101) consists of a single element, namely f:r! ,, E S(sE*). 113) are equivalent to Re fx-o (g 0 -g) 0 >0 (gEG). 4. 9. Jj. and g0 EG. The following statements are equivalent : 1° g 0E'i£G(x). 114) where ~ is a scalar such that I ~ I = 1.
28). e. 29), which completes the proof. Let us consider now the problem of simultaneous characterization of a set _M cG of elements of best approximation. [J.! ~~ (q) ,. L! 31). =F 1 taking (qE V), we would 32 Approximation by elements of arbitrary linear subspaces Chap. 4. v! C G. We have M C ~a( x) if and only if there exists a Radon measure [1. on Q satisfying ( 1. 25), ( 1. 27) and x(q) -g0 (q) =[sign~(q)]max]x(t)-g 0 (t)l d][L] (qES([L), g0 EM). 33) Proof. Assume that we have Me ~a(x). 3, a Radon measure [1.