A Recent Application of Enflo’s BAP Result

The bounded approximation property (BAP) is a central concept in Banach space theory. The existence of Banach spaces lacking this property, long an open question, was first established by Per Enflo in 1973. In this work, we consider Banach spaces of holomorphic functions on the unit disk and introduce the notion of a linear polynomial approximation scheme (LPAS), namely, a sequence of bounded linear operators mapping the space into itself such that, for each function, the resulting sequence of polynomials converges in norm. We show that a complete characterization of spaces admitting such schemes naturally involves the BAP. In particular, and somewhat surprisingly, the mere density of polynomials in the space does not guarantee the existence of a polynomial approximation scheme; the space must also satisfy the bounded approximation property.