# Schauder lemma

The following theorem is in the functional analysis literature generally referred to as the Schauder lemma. It is a version of the open mapping theorem in Fréchet spaces and is often used to verify the open-ness of linear, continuous maps.

Theorem. Let $E,F$ be Fréchet spaces. Denote by $\mathcal{U}_{0}(E),\mathcal{U}_{0}(F)$ the zero neighborhood filter of $E$ and $F$ respectively. Let $T\colon E\to F$ be a linear and continuous map which is almost open, i.e.

 $\displaystyle\forall_{U\in\mathscr{U}_{0}(E)}\overline{T(U)}^{F}$ $\displaystyle\in\mathscr{U}_{0}(F)$

Then $T$ is open.

Title Schauder lemma SchauderLemma 2013-03-22 19:09:44 2013-03-22 19:09:44 karstenb (16623) karstenb (16623) 4 karstenb (16623) Theorem msc 54E50 msc 46A30 OpenMappingTheorem almost open set