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In this paper we consider the problem of best approximation in p (N), 1 < p ∞. If h p , 1 < p < ∞ denotes the best p -approximation of the element h ∈ 1 (N) from a finite-dimensional affine subspacewhere h * ∞ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to give a complete description of the rate of convergence of h p − h * ∞ as p → ∞ by proving that there are constants L 1 , L 2 > 0 and 0 a 1 such thatfor p large enough. (J.M. Quesada), jmmoreno@ujaen.es (J. Martínez-Moreno), jbusta@fcm.uap.mx (J. Bustamante).

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Let h p , 1 < p < ∞, be the best p -approximation of the element h ∈ R n from a proper affine subspace K of R n , h / ∈ K, and let h * ∞ denote the strict uniform approximation of h from K. We prove that there are a vector ∈ R n \{0} and a real number a, 0 a 1, such that h p