¤ϕ.
Thus, recalling that
Γid(u, v) =−A−1
uv+12uxvx
x, we arrive by right invariance of Γ at
(∇XY)(ϕ) =DY(ϕ)·X(ϕ)−Γϕ(Y(ϕ), X(ϕ)).
A. Differential calculus in Banach spaces
For Banach spacesEandFwe letL(E,F) be the Banach space of continuous linear mapsE!F. For anyk≥1,Lk(E;F) denotes the Banach space of continuous k-multilinear maps E!F, and Lksym(E;F)⊂Lk(E;F) is the subset of symmetric maps. For a Banach manifoldM,Lksym(TM;R) denotes the vector bundle overM with fiber Lksym(TmM;R) overm∈M.
Let U be an open subset of E. As usual when dealing with Banach spaces a continuous mapf:U!Fis said to beC1ifDf: U!L(E,F) is continuous. Since L(E,F) is a Banach space we may defineDpf recursively for anyp≥1. IfGis also a Banach spaces and f a map (U ×V ⊂E×F)!G, we writeD1f:U ×V!L(E,G) for the partial derivative with respect to the first variable.
We will also need a different notion of differentiability. Recall that a Fr´echet space is a complete Hausdorff metrizable locally convex topological vector space.
DefinitionA.1. Let E and F be Fr´echet spaces, U be an open subset of E, and f:U!F be a continuous mapping. We say that f is Gateaux-C1 if for each
pointx∈U there exists a linear mapDf(x) :E!F such that Df(x)v= lim
t!0
f(x+tv)−f(x) t for allv∈E and the map
(x, v)−!Df(x)v: U ×E−!F (A.1)
is continuous (jointly as a function on a subset of the product).
We say that f is Gateaux-C2 if both f and the map in (A.1) are Gateaux-C1. The notion of Gateaux-Cp, p≥3, is defined inductively. See [15] for an extensive presentation of calculus for Gateaux differentiable functions in Fr´echet spaces.
Clearly, iff is a map between Banach spaces so that both definitions of dif-ferentiability apply, then normal differentiability implies Gateaux differentiability, that is, if f is Cp, p≥1, then f is Gateaux-Cp. The basic result in the converse direction is the following.
Proposition A.2. Let E andFbe Banach spaces,U be an open subset of E, andf:U!Fbe a continuous mapping. Iff is Gateaux-Cp+1 for some p≥0, then f isCp. In particular, for smooth maps between Banach spaces the two definitions coincide.
Proof. We refer to the book by Keller [19, p. 99 (see also the remark on p. 110)].
Note that our Gateaux-Cp maps correspond to the classCcp in [19] . We also have the following result.
Proposition A.3. ([26, Theorem 5.3]) Let E, F and G be Banach spaces, and let U be an open subset of E. Let f be a Cp-mapping of U ×F into Gsuch that f(x, u) is linear with respect to the second variable u. Seth(x)u=f(x, u)and regard has a mapping ofU intoL(F,G). Thenhis aCp−1-mapping.
The typical application in this paper of Propositions A.2 and A.3 is the fol-lowing consequence. Let E and F be Banach spaces, U be an open subset of E, and (x, u, v)!P(x, u, v) :U ×E×E!Fbe a continuous mapping linear inuandv.
Suppose we can show that P is Gateaux-Cp+1. ThenP is Cp by Proposition A.2.
Hence Proposition A.3 shows that the map
(x, u)−!(v!P(x, u, v)) :U ×E−!L(E,F),
isCp−1. Using thatL(E, L(E,F))L2(E;F),another application of Proposition A.3 yields that
x−!((u, v)!P(x, u, v)) :U ×E−!L2(E;F)
is Cp−2. The upshot is that if P is Gateaux-smooth then x!Px=P(x,·,·) is a smooth map U!L2(E;F).
We also need the following corollary of the inverse function theorem.
Proposition A.4. ([21, Proposition 5.3]) Let U and V be open subsets of Banach spaces and let f: U!V be a Cp-map which is also a C1-diffeomorphism.
Then f is a Cp-diffeomorphism.
Acknowledgements. The author is grateful to Professor Boris Kolev and the referee for valuable suggestions. The research presented in this paper was carried out while the author visited the Mittag-Leffler Institute in Stockholm, Sweden.
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Jonatan Lenells
Department of Mathematics Lund University
P.O. Box 118 SE-22100 Lund Sweden
jonatan@maths.lth.se Received December 16, 2005 published online May 22, 2007