Schauder lemma

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

Theorem. Let E,F be Fréchet spaces. Denote by 𝒰0(E),𝒰0(F) the zero neighborhood filter of E and F respectively. Let T:EF be a linear and continuous map which is almost open, i.e.

U𝒰0(E)T(U)¯F 𝒰0(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