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 ${𝒰}_0(E),{𝒰}_0(F)$ the zero neighborhood filter of $E$ and $F$ respectively. Let $T:E\to F$ be a linear and continuous map which is almost open, i.e.

${\forall }_{U\in {𝒰}_0(E)}{\overline{T(U)}}^F$$\in {𝒰}_0(F)$

Then $T$ is open.