High Quality Content by WIKIPEDIA articles! In mathematics, there are two theorems with the name "open mapping theorem". In both cases, they give conditions under which certain maps are open maps, i.e. they map open sets to open sets. They are significant results in their respective contexts since, unlike inverse images, direct images of functions are much less tractable in general. In functional analysis, the open mapping theorem or Banach–Schauder theorem states that a surjective continuous linear transformation of a Banach space X onto a Banach space Y is an open mapping. In complex analysis, the open mapping theorem states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping.

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