Most spaces considered in functional analysis have infinite dimensionsdimension. To show the existence of a [[vector space basis]] for such spaces, it may require [[Zorn's lemma]]. However, a somewhat different concept, the [[Schauder basis]], is usually more relevant in functional analysis. Many very important theorems require the [[Hahn–Banach theorem]], which is usually proved using the [[axiom of choice]], although the strictly weaker [[Boolean prime ideal theorem]] suffices. The [[Baire category theorem]], which is needed to prove many important theorems, also requires a form of axiom of choice.
