On a theorem of Lyusternik
We propose a new proof of the following Lyusternik's theorem. Let E,F be Banach spaces, \(X\subset E\) be an open set and let \(x_ 0\in X\). If \(f: X\to F\) is a map strictly differentiable at \(x_ 0\) and if the differential \(Df(x_ 0)\) of f at \(x_ 0\) maps E onto F, then there exists a constant \(M>0\) and a neighborhood U of \(x_ 0\) in X such that \(dist(x,\Omega)\leq M\| f(x)- f(x_ 0)\|\) for all \(x\in U\), where \(\Omega =\{y\in X:\quad f(y)=f(x_ 0)\}.\)
scientific article; zbMATH DE number 3929711