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

By Prof. Ivan Singer (auth.)

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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 ([198], theorem 1). 30). **) See N. Bourbaki [22], 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.

J. 6). In the particular case when T = = [0, 1], v = the Lebesgue measure and the scalars are real, the equivalence 1°~3° was given by V. N. Nikolsky ([169], p. 106, formula (1)) and in the general case by B. R. Kripke and T. J. 4); the above proofs of the equivalences 1°~3°~ 4° have been given, essentially, in the paper [232], p. 353-354. 7. Let E = L 1 (T,v), where (T,v) is a positive measure space with the property that the dual L 1 ( T, v)* is canonically equivalent to L"" (T,v), and let G be a linear subspace of E, x e E"' Gand y0 e ~G(x).

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'

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