Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Abstract

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x 0)⊂E→F be C1 nonlinear map, whereU (x 0) is an open set containing pointx 0∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x 0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.