# Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

By A. I. Kostrikin, I. R. Shafarevich

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2 (The converse of Taylor's theorem). Let f : U -p F be a mapping. 2) f (x +y) = f (x) + Z Pz(x)(y, ... ,y) ' o(uIyIuc) I. INFINITE-DIMENSIONAL CALCULUS 20 for every x e U and for every y with sufficiently small Ilyll, then f is a C'`-mapping and cpi = (df). 2). Let U be an open subset of a CLCTV-space E. A mapping f of U into another CLCTV-space F is called a C°-mapping if it is continuous. 3) F(v) =f(x+v) -,f(x) - (Df)(x)(v) -... _ (Dnf)(x)(v,... 4) r F(tv)/t'', 0, t # 0, t=0 is continuous on a neighborhood of (0,0) E R x E.

1) d c(t) = v(c(t)), c(0) = x is called the integral curve of v with initial point X. If v satisfies the Lipschitz condition, then the existence and the uniqueness of the integral curve are well known and the integral curve is continuous with respect to the initial value X. 1 (Flow box theorem). Let v : U - E be a CT (1 0) and a CT mapping P : V x (-S, S) - U, called an integral mapping or the flow of v, such that (a) cp is uniquely determined by v, (b) for each x e V, the curve cow (t) defined by cow (t) = co(x, t) is the integral curve of v with the initial point x, (c) for each t e (-S, S), the mapping cot defined by cot (x) = co(x, t) is a CT diffeomorphism of V into U, (d) every integral curve of v with initial point x e V coincides with cow (t) whenever t is contained in the defining domains of both integral curves.

However, we do not use this condition in what follows. For xo E U fl E, let Wo be a convex neighborhood of xo contained in the unit open ball in Ed with the center at xo . 2) holds on Wo fl E. 1 can be applied to the C°° vector field on Wo. 1 are satisfied. 5. 1. 3. PROOF. 1. (a) and (c) are trivial. We shall prove (b) below. cot is a C°° diffeomorphism of V onto cot (V) (C Wo). To prove that cot V fl E'c -* co(V) fl E' is a C°° diffeomorphism, it is enough to prove that Pt 1 co(V) fl E' -* V fl E'c is a C°° mapping.