On the domain of implicit functions in a projective limit setting without additional norm estimates (Q2193993)

The author proves an implicit function theorem without metric or norm estimates in the category of Fréchet spaces which are considered as projective limits of Banach spaces. Let \(E_{\infty} = \varprojlim E_i\) and \(F_{\infty} = \varprojlim F_i\) be Fréchet spaces that are projective limits of the sequences of Banach spaces \((E_i)\) and \((F_i)\), respectively. Let \(U_0\) and \(V_0\) be open neighborhoods of zero in \(E_0\) and \(F_0\), respectively. Let \(f : U_0 \times V_0 \to F_0\) be a Fréchet differentiable function of class \(c^r, r \geq 1\), such that \(f(0,0)=0\) and its differential with respect to the variable in \(F\) at \((0,0)\) is the identity, \(D_2f(0,0)= Id_F\). Suppose \(U_{\infty} = E_{\infty} \cap U_0, V_{\infty} = F_{\infty} \cap V_0\), and \(f_{\infty} = \varprojlim f_i\), then there exist a domain \( \emptyset \neq D \subset U_{\infty} \) ( not necessarily open and it contains the unit ball of some Banach space) and a function \(u_{\infty}: D_{\infty} \to V_{\infty}\) such that \[ f_{\infty}(x, u_{\infty}(x)) =0 \forall x \in D_{\infty}. \] The author applies this theorem to obtain the corresponding inverse function theorem in the same framework. \newline It is worth to mention that since domains can possibly be non-open, the regularity results on implicit and inverse functions can not be stated and this imposes an obstruction to adapt the Fröbenius theorem which requires differentiation on domains.