On the Role of ℓ_∞ in Approximation Theory

Abstract

Given a family of finite-dimensional subspaces V _n in C[0,1], it is important to develop an algorithm that to a given function f ∈ C[0,1] assigns an approximation P _n f ∈ V _n . This algorithm should be “ simple ” and “ good ”. That translates into P _n being a continuous linear map from C[0,1] in V _n such that ∥f − P _n f∥ ≤ C dist(f, V _n ) where the constant C does not depend on n. Using the principle of uniform boundedness it is easy to show that the above-mentioned conditions are equivalent to the existence of projections (linear, idempotent operators) P _n from C[0,1] onto V _n such that ∥P _n∥ ≤ C.