In this paper we study some approximation theory questions which arise from the analysis of the discretization error associated with the use of the Rayleigh-Ritz-Galerkin method for approximating the solutions to various of boundary value problems, cf. [13, [2], [33, [43, [7], [8], [93, [12], [143, [18], [19], [20] and [22]. In particular, we consider upper and lower bounds for the error in approximation of certain families of functions in Sobolev spaces, cf. [15], by functions in finite-dimensional polynomial spline types subspaces, cf. [16]. In doing this, we directly generalize, improve, and extend the corresponding results of[1], [17], [18], [19], [20], and [21]. Throughout this paper, the symbol K will be used repeatedly to denote a positive constant, not necessarily the same at each occurrence and the symbol μ will be used repeatedly to denote a nonnegative, continuous function on [0,∞], not necessarily the same at each occurrence.