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: