We present a way to organize a constructive development of the theory of
Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We
illustrate this by giving elementary proofs of Wiener's result on the inverse
of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we
show how this can be used in a localic, or point-free, description of the
spectrum of a Banach algebra
