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
Canonical name	SchauderLemma
Date of creation	2013-03-22 19:09:44
Last modified on	2013-03-22 19:09:44
Owner	karstenb (16623)
Last modified by	karstenb (16623)
Numerical id	4
Author	karstenb (16623)
Entry type	Theorem
Classification	msc 54E50
Classification	msc 46A30
Related topic	OpenMappingTheorem
Defines	almost open set