`A direct proof of Wiener's theorem'

In functional analysis it is not uncommon for a proof to proceed by contradiction coupled with an invocation of Zorn's lemma. Any object produced by such an application of Zorn's lemma does not in fact exist, and it is likely that the use of Zorn's lemma is artificial. It has turned out that many proofs of this sort can be simplified, both in form and complexity, with the principle of open induction isolated by Raoult as a substitute for Zorn's lemma. If moreover the theorem under consideration is sufficiently concrete, then a far weaker instance of induction suffices and, with some massaging, one may obtain a fully constructive proof. In the present note we apply this method to Gelfand's proof of Wiener's theorem, producing first a simple direct proof of Wiener's theorem, and then an even simpler constructive proof. With this example in mind we look toward developing a more generally applicable technique.