Suggested by Shepp [Ann. Math. Statist. 36(4) (1965), 1107-1112] we defined a sequence space Λ_p(f) determined by a single function f(≠ 0) ∈ L_p(R, dx), 1 ≤ p < +∞, and discussed the structure of it. The problems are the linearity and the visible sequential representation of Λ_p(f). In this paper we name Λ_p(f) a Shepp space and discuss the problems in the case of p = 2 by defining an inner approximation Λ⁰₂(f) and an outer approximation Λ^φ₂(f) of Λ₂(f), and we give a necessary and sufficient condition for Λ⁰₂(f) = Λ^φ₂(f) in terms of doubling dimension. In this case Λ₂(f) is a linear space and those approximationsare its visible sequential representations. We also give an example such that Λ₂(f) is a linear space but Λ⁰₂(f) ≠ Λ^φ₂(f).