Complete rank theorem of advanced calculus and singularities of bounded linear operators
The article deals with generalizations of classical results in calculus about smooth systems of functions in several variables the local representation of a function \(f: {\mathbb R}^m \to {\mathbb R}^n\), \(f \in C^r\), in the form \[ f(x) = (\psi \circ f'(x_0) \circ \varphi)(x),\tag{1} \] where \(\varphi, \psi(x) \in C^r\), \(\varphi(x_0) = 0\), \(\varphi'(x_0) = I\), \(\psi(0) = f(x_0)\), \(\psi'(0) = I\). The author remarks that this representation is true if \(f \in C^r\) is a mapping between Banach spaces \(E\) and \(F\) and \(\text{ker} \, f'(x_0)\) and \(\text{im} \, f'(x_0)\) are complemented subspaces in \(E\) and \(F\) correspondingly. The proof repeats the standard arguments. In connection with this theorem, the author repeats standard theorems about the stability properties of linear Fredholm and semi-Fredholm operators, some results about the existence of generalized inverse operators, and so forth.
scientific article; zbMATH DE number 5322575