| Given a reflexive Banach space X .In the ring of C∞ function germ s∈(X) ,any real singular germ f in ∈(X) whose second Frechet derivative at origin f″(0)=T is a Fredholm operator is isomorphic to a germ of the form 1/2+g(v) .If we replace g by a g１ which is isomorphic to g ,we clearly obtain a germ in ∈(X) which is isomorphic to the original one. However,is true converse of this proposition?In this paper,we will show that the converse is also true. |
