P is a homomorphism of A-modules, we have TS(v 0 u) = TS(v)o TS(u). , TS(N) - - - T(N) is commutative, where the horizontal arrows denote canonical injections.

Ii) Let x = (Xi)i E I be a family of elements ofE satisfying a) and b) of (i). Then there exists one and only one unital continuous homomorphism

---+U«Xi)iEI) of A[(Xi)iEI] into E, and let V be a neighbourhood of 0 of E which is an ideal of E.

Since the ring K[X] is a subring of K[[X]], every rational fraction u/v E K(X) (u, v being polynomials in X) may be identified with the (generalized) formal power series uv- 1 of K«X», which we shall call its expansion at the origin; the field K (X) is thus identified with a subfield of K «X». 10. Exponential and logarithm By the exponential power series we shall understand the element n~O Q[[X]] ; it will be denoted by exp X or eX. 13. - In Q[[X, Y]] we have e X + Y = eXe Y . For the binomial formula gives PROPOSITION (X+ Yt n!