A NOTE ON THE INVERSE FUNCTION THEOREM OF NASH AND MOSER

The Nash-Moser inverse function theorem is proved for different kind of differentiabilities.

possible to derive the inverse function theorem of Nash and Moser for natural C differentiabilities stronger than the differentiability C 2. THE INVERSE FUNCTION THEOREM OF NASH AND MOSER.
Let LC denote the category of locally convex limit vector spaces [2] and con- tinuous linear mappings.Further let K denote a coreflective subcategory of LC which is closed under finite products and the coreflector .LC K s is the identity on morphisms and such that the identity mapping (Cc(X F)) C (X,F) C (X F) is C endowed with continuous convergence [2], and X is a limit space and F E obj(LC) For any pair E,F E obj(LC) we let Lk(E F) be the space of all continuous C k-linear mappings from E k into F endowed with continuous convergence.We write (ekc(E F)) ek(E,F) Dkf(x)) Dk+|f(x)h t+0 and such that for each k IN k < p the following two conditions are satisfied: By Keller [2] the chain rule is valid for Coo since a is a finer limit ( structure than continuous convergence.From the universal property of continuous con- vergence follows that for any continuous map g U-Lk(E,F) the associated map

Ux
/F defined by (x,hl tinuous.As the limit structure is always finer than c we have that differenti- The latter is exactly ability of class C a implies differentiability of class C c the cDncept of differentiability used by Hamilton [I] to prove the inverse function theorem of hash and Moser.
We first recall some definitions that will be needed.
Let E be a Frchet space.A grading on E is an increasing sequence of norms (li'II r)riN on E which defines the topology on E Two gradings (II'II r rEIN < cllxli2 and and II'112r)rCIN are equivalent if for some s IN llXllr r+s II..x.. r2 _< c llx llr+s2 X E with a constant c which may depend on r A graded space is a Frchet space together with an equivalence class of gradings.We say that a graded space E admits smoothing operators if we can find linear maps S t E such that for some r liSt(x) lli+k < ctr+kllxll i and llSt(x) xll i < ctr-kllxlli+k for all i,k f IN < t < Oo, x E and some constant c which may depend on i and k.
Let E and F be graded spaces and U open in E We say that a map f U-F is tame if for every x E U we can find a neighbourhood U U, where the constant c may depend on n In the proof of the inverse function theorem of hash and Moser we shall also need the following result (Lemma 2, [3]): The composition of two continuous tame maps is continuous and tame.
THEOREM.Let E and F be graded spaces which admit smoothing operators.Let   U be open in E and assume that (I) f U-F is differentiable of class C and tame.
(2) DKf U Ek-F is tame for every k IN.
(3) For each x f U the derivative Df(x) E F is an isomorphism.and assumption (2).Further the assumptions (4) and (5) imply that also Vf Ux F E is continuous and tame.Now we have that f is differentiable of class C a and all Dkf are tame, Df(x) E -F is an isomorphism for every x t U and the family of inverses Vf U F E are continuous and tame maps.Consequently the conditions of the inverse function theorem of Nash-Moser are fulfilled (theorem 1.1.1p. 171 in [I]).Then for every x E U there exist neighbourhoods U of x V is bijective and f V U and V of f(x 0) such that f U is continuous and tame.Furthermore the formula lim t-1(f-1(y + tw) f-1(y)) Vf(f-1(y))w holds, From the definition of the c-differentiability follows that the map f- Dk+If -I is clearly tame so we only have to show that Vf is differentiable of class C k By induction on p By theorem 5.3.1, p. 102 in [I] we have that Vf is weak- ly differentiable and that D) U E F E is continuous and the formula [D(Vf)](x){u,w} -Vf(x)[D2f(x){u,Vf(x)w}] holds for all x U u E E and w F. Thus the derivative D(Vf) U Lc(E F,E) can be factorized according to (D2f--) e (E,F) L((F,E) e (E F,E)

Uo c oc
where h is defined by h(,) o(idE, ) for D2f(x) and Vf(x) By theorem 0.3.5 in [2] h is continuous for c Since the category K is closed under finite products and ? is a coreflector it follows that h is continuous.
Thus it is true for p Since h is bilinear it is differentiable of class C a and consequently the map Vf is differentiable of class C a by induction Thus the theorem is proved We shall now consider examples of coreflective subcategories of LC which are closed under finite products and the coreflectors fulfill the assumption that the identity mapping Cc(U F) C (U,F) for some filter G which converges to zero in E EXAMPLE 3. Let K be the category KM of Marinescu spaces [2].The corflector M LC KM is the identity on morphisms and on objects E it is characterized as follows: a filter F on E converges to zero in E M iff G G < F and N{G G E } E for some filter which converges to zero in E EXAMPLE 4. Let be the category b of bornological locally convex limit b Kb vector spaces.The coreflector LC is the identity on morphisms and on objects E it is characterized as follows: a filter F on E converges to zero in E b iff B < for some bounded subset B c E i.e. some set B such that VB converges to zero in E Example gives us the inverse function theorem of Nash and Moser by Hamilton [|].
From example 3 we derive the inverse function theorem of Nash and Moser for the differentiability of class C M (C A in Keller [2]).In [4] Kriegl has discussed smooth mappings between locally convex spaces, where a mapping is called smooth iff its composition with smooth curves are smooth.He has compared this concept of smooth- ness with different C -differentiabilities (see [2]).From [2] and [4] follow that a mapping between Frchet spaces is smooth iff it is C -differentiable.Thus the inverse c function theorem of Nash and Moser is valid for this concept of smoothness.
such that for every n C IN we have the growth estimate f(x)II n n+r all x

( 4 )
The map Vf U L (F,E)Vf(x) (Df(x)) -I is continuous.(5) Vf U xF E is tame.Then for any x E U we can find open neighbourhoods of x for all k t IN Furthermore we have the formula D(f-1)(y) Vf(f-l(y)) for all y V PROOF.The maps U Ek F are continuous and tame, since f is differenti- able of class C DEFINITION.Let E and F be locally convex spaces and let U be open in E A mapping f U F is said to be differentiable of class C p if there exist is continuous.