A note on inverse function theorems

In this note we use the Newton-Raphson approach to inverse function theorems. We draw natural conclusions when only a left or a right inverse to the differential at a point is available. By using a strengthened version of differential, we are able to use differentiability at a single point as the smoothness condition. Although the method has been used before (cf. 2, p. 167 ff.), analysis books have tended to use an approach that assumes finite dimensionality of the reference spaces. DEFINITION. Let U and V be Banach spaces and f: U-* V a function. A strong differential of f at a point xO in U is a bounded linear transformation a: U-* V which approximates changes of f in the sense that for every E> 0, there is a 5 >0 such that if x' and x are nearer than a to 0, then: