arXiv:1803.09874v1 [math.FA] 27 Mar 2018 BERNSTEIN LETHARGY THEOREM AND REFLEXIVITY ASUMAN G ¨ UVEN AKSOY * , QIDI PENG 1 Abstract. In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein’s Lethargy Theorem. Let X be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence {d n } n≥1 decrease to 0. Suppose that {Y n } n≥1 is a system of strictly nested subspaces of X such that Y n ⊂ Y n+1 for all n ≥ 1 and for each n ≥ 1, there exists y n ∈ Y n+1 \Y n such that the distance ρ(y n ,Y n ) from y n to the subspace Y n satisfies ρ(y n ,Y n )= ‖y n ‖. Then, there exists an element x ∈ X such that ρ(x, Y n )= d n for all n ≥ 1. 1. Introduction Our work is motivated by one of the notable theorems used in the constructive theory of functions, termed the Bernstein’s Lethargy Theorem (BLT) [11]. For a subspace A of a normed linear space (X, ‖·‖), we define the distance from an element x ∈ X to A by ρ(x, A) := inf {‖x − a‖ : a ∈ A}. The Weierstrass Approximation Theorem states that polynomials are dense in C [0, 1], thus it is known that for any f ∈ C [0, 1], lim n→∞ ρ(f, Π n )=0, where for each n ≥ 1, Π n is the vector space of all real polynomials of degree at most equal to n. However, the Weierstrass Approximation Theorem gives no information about the speed of convergence for ρ(f, Π n ). In 1938, S.N. Bernstein [11] proved that if {d n } n≥1 is a non-increasing sequence of positive numbers with lim n→∞ d n = 0, then there exists a function f ∈ C [0, 1] such that ρ(f, Π n )= d n , for all n ≥ 1. This remarkable result is called the Bernstein’s Lethargy Theorem (BLT) and is used in the constructive theory of functions [28], and in the theory of quasi- analytic functions in several complex variables [24]. More generally, let {Y n } be a system of strictly nested subspaces of the Banach space X . The sequence of errors of the best approximation from x ∈ X to Y n , denoted by {ρ(x, Y n )}, may converge to zero at an arbitrarily slow rate or an arbitrarily fast rate. For example, Shapiro in [26], replacing C [0, 1] with 2010 Mathematics Subject Classification. Primary 41A25; Secondary 41A50, 46B20. Key words and phrases. Best approximation, Bernstein’s lethargy theorem, reflexive Banach space, Hahn–Banach theorem. 1