On the Banach Differential Structure for Sets of Maps on Non-Compact Domains

Paolo Piccione^{1} and Daniel V. Tausk^{2}

Departamento de Matemática

Universidade de São Paulo, Brazil

Abstract. We study the differentiable structure for certain sets of maps between a topological space M and a differentiable manifold N. We axiomatize two properties for the Banach space of functions 𝔐(M, IR) that guarantee the existence of an intrinsic Banach differentiable structure for the set of maps 𝔐(M, N). The case of C_{b}(M, N) of all bounded continuous functions from M to N, and the case of C_{0}(M, N) of functions vanishing at infinity are used as basic examples for the theory. Some examples are given, in particular it is discussed the case of Sobolev spaces of mixed type.

1. Introduction

The aim of this paper is to introduce formally a Banach differentiable structure for certain sets of maps 𝔐(M, N) between a topological space M and a differentiable manifold N. Our assumptions on the base space are rather weak; the target manifold N is assumed to have sufficient regularity (C^{k}, k ≥ 3) and to be embedded as a submanifold of IR^{n}. Moreover, we show that the Banach differential structure of 𝔐(M, N) does not depend on the embedding of N.

Such a problem, in the case of a non compact base space M, has not been treated satisfactorily in the classical literature (see [3,6]). We remark that the classical technique of Vector Bundle Neighborhoods used in [6] is not applicable in the case of a non compact domain. The main reason for this is that there is no natural notion of bounded sections of a vector bundle over non compact domain, unless one uses some extra structure on the vector bundle, such as a smoothly varying inner product on the fibers. On the other hand, the introduction of such extra structure would cause that the transition functions between two

1 Partially sponsored by CNPq, Processo n. 301410/95-0 2 Sponsored by FAPESP overlapping charts (given by composition maps) are in general non continuous with respect to the induced metric.

It is highly desirable to extend some of the results of [6] to the non compact base space, due to possible applications to Calculus of Variations, Critical Point Theory and Global Analysis on manifolds.

Among possible applications, we mention the study of harmonic maps between two Riemannian manifolds, where general existence questions arise when the domain M is non-compact (see [4] and the references therein).

The initial motivation for this paper is a recent work by Benci, Giannoni and Piccione (see [1]), who studied local minima of a functional defined on maps from IR^{3} to a compact Riemannian manifold M. The solutions to this variational problem have applications to field theory and they are related to the existence of topological solitons.

Given a topological space M and a Banach space 𝔐(M, IR) of real valued functions on M, we axiomatize two properties (axioms (Ax1) and (Ax2) introduced in Section 4) that allow to define the set 𝔐(M, N) for every differentiable manifold N with sufficient regularity and to prove the existence of an intrinsic Banach differentiable structure on 𝔐(M, N). The two axioms (Ax1) and (Ax2) are basically contained in the Axiom (B §5) introduced in [6]

The abstract theory is built on two basic models: the set C_{b}(M, N) of continuous functions from M to N having precompact image in N, and the set C_{0}(M, N) of continuous functions vanishing at infinity.

The main tool used in the construction of the differentiable structure for 𝔐(M, N) is an infinite dimensional extension of the Rank Theorem, that gives the local form of those maps having differential with image of constant dimension. Then, if N is immersed in some Euclidean space IR^{n}, the differentiable structure for 𝔐(M, N) is obtained as a submanifold structure of the Banach space 𝔐(M, IR^{n}).

The paper is organized as follows.

In Section 2 we state and prove some abstract results on maps between Banach manifolds having constant rank. In particular, we derive the local form of a map with constant rank on an open set (Proposition 2.3) and we prove the submanifold structure for the image of an open set for and the inverse image of a point through a map having constant rank (Corollaries 2.5, 2.6 and 2.7).

In Section 3, as an example for our theory, we discuss the case of C_{b}(M, N), the set of continuous functions from M to N having precompact image in N.

In Section 4 we generalize the results of Section 3 to a more general case. We state two axioms for the Banach space 𝔐(M, IR), for a fixed topological space M, that guarantee the existence of a differentiable structure for 𝔐(M, N) for every manifold N which is embedded in some IR^{n} (Theorem 4.2). Moreover, we prove the functoriality for 𝔐, and, in particular, we prove that if the two axioms are satisfied, then the differentiable structure for 𝔐(M, N) does not depend on the embedding of N (Corollary 4.4).

In Section 5 we discuss the case of C_{0}(M, N), the set of continuous functions from M to N vanishing at infinity and, more in general, the case of a Banach space of functions 𝔐_{0}(M, IR) ⊂ C_{0}(M, IR). The space C_{0}(M, IR) does not satisfy the second axiom introduced in Section 4, and thus one needs to modify properly the axioms in order to extend the results of the previous section to this large class of examples.

Finally, in Section 6 we consider some concrete examples of our theory. The main emphasis is given to a mixed Sobolev spaces H_{p,q}(M, N) of maps between a Riemannian manifold M and a differentiable manifold N.

Acknowledgements: The authors wish to express their gratitude to Prof. Franco Mercuri, Prof. Vieri Benci and Prof. Fabio Giannoni for many fruitful conversations about the problem.

2. The Rank Theorem in Infinite Dimension

In this section we state and prove an infinite dimensional extension of the rank theorem, that gives a local characterization of those functions between open subsets of Banach spaces having derivative of constant rank. The techniques used are elementary; we claim no originality for most of the results presented.

Let E and F denote two real Banach spaces. If f : U ⟼ F is a map of class C^{k}, k ≥ 1, between open subset U ⊆ E and F, for z ∈ U we denote by df(z) its derivative, which is a linear bounded operator between E and F. If T is any linear map from E to F, Im(T) and Ker(T) will denote respectively the image and the kernel of T.

We will say that a closed subspace S of a Banach space Ẽ is complemented in Ẽ if there exists another closed subspace C of Ẽ such that S ⊕ C = Ẽ, i.e., if S ∩ C = {0} and S + C = Ẽ. It is an easy observation that, if Ẽ is a finite dimensional space, then two subspaces S_{1} and S_{2} have a common complement C if and only if they have the same dimension.

We denote by L(E, F) the set of all continuous linear maps from E to F endowed with the operator norm topology; by In(E, F) the maps that are injective and have image which is closed and complemented in F, and by Sub(E, F) the set of those maps which are surjective and with kernel complemented in E.

A C^{1}-map f : U ⊂ E ⟼ F is said to be an immersion at x ∈ U if df(x) ∈ In(E, F); f is said to be a sub-mersion at x if df(x) ∈ Sub(E, F).

It is not difficult to prove that In(E, F) and Sub(E, F) are open subsets of L(E, F).

We recall the following local characterization of immersions and sub-mersions, whose proof can be found in [5]:

Proposition 2.1. Let f : U ⟼ F be a map of class C^{k}, k ≥ 1, and let z_{0} be a point in U.

(1) If f is an immersion at z_{0}, then, given any closed complement F_{1} of Im(df(z_{0})) in F, there exists a neighborhood V of f(z_{0}) in F, a neighborhood V_{1} of z_{0} in U, a neighborhood V_{2} of 0 in F_{1} and a diffeomorphism β : V ⟼ V_{1} × V_{2} of class C^{k} such that β(f(z_{0})) = (z_{0}, 0), and the composition (β ∘ f) : V_{1} ⟼ V_{1} × V_{2} is given by:

(β ∘ f)(x) = (x, 0), ∀ x ∈ V_{1}.

(2) If f is a sub-mersion at z_{0}, then there exists an open neighborhood Z of z_{0} in U, an open neighborhood V_{1} of f(z_{0}) in F, an open neighborhood V_{2} of 0 in Ker(df(z_{0})) and a C^{k}-diffeomorphism α : V_{1} × V_{2} ⟼ Z such that α(f(z_{0}), 0) = z_{0} and the composition (f ∘ α) is given by:

(f ∘ α)(x_{1}, x_{2}) = x_{1}, ∀ x_{1} ∈ V_{1}, x_{2} ∈ V_{2}.

We give the following definition: Definition 2.2. A map f : U ⟼ F of class C^{k}, k ≥ 1, is said to have constant rank in U if:

(a) Ker(df(z)), Im(df(z)) are closed and complemented for all z ∈ U;

(b) there exists a closed subspace S of F which is a complement for all the subspaces Im(df(z)) in F, z ∈ U.

It is easily seen that, if f has constant rank in U and z_{0} ∈ U, then, given a common complement F_{1} for all the subspaces Im(df(z)), z ∈ U and denoted by π the projection

π : Im(df(z_{0})) ⊕ F_{1} ⟼ Im(df(z_{0})),

then the restriction π|_{Im(df(z))} : Im(df(z)) ⟼ Im(df(z_{0})) is an isomorphism for all z ∈ U.

If X and Y are two Banach manifolds and f : X ⟼ Y is a C^{1}-map, then f is said to have locally constant rank if around every x ∈ X there is a local representation of f in coordinates which has constant rank. Observe that, for a local representation of f in coordinates, the property of having constant rank may depend on the choice of the coordinate system.

We prove the following local characterization for maps of constant rank:

Proposition 2.3. Let f : U ⊂ E ⟼ F be a map of class C^{k}, k ≥ 1, having constant rank in the open set U. Let z_{0} be a point of U and set E_{1} = Im(df(z_{0})) and E_{2} = Ker(df(z_{0})); moreover, let F_{1} be a closed subspace of F which is a common complement for all the subspaces Im(df(z)), z ∈ U. Then, there exist an open neighborhood Z of z_{0} in U, an open neighborhood W of f(z_{0}) in F, a C^{k}-diffeomorphism α : A ⟼ Z, where A is an open neighborhood of 0 in E_{1} ⊕ E_{2}, a C^{k}-diffeomorphism β : W ⟼ B, where B is an open neighborhood of 0 in E_{1} ⊕ F_{1} such that α(0, 0) = z_{0}, β(f(z_{0})) = (0, 0), and the composite map (β ∘ f ∘ α) is given by:

(β ∘ f ∘ α)(x_{1}, x_{2}) = (x_{1}, 0), ∀ (x_{1}, x_{2}) ∈ A.

Proof: We will assume without loss of generality that f(z_{0}) = 0 ∈ E_{1}.

Let π : F = E_{1}⊕F_{1} ⟼ E_{1} be the projection onto the first summand; observe that, since F_{1} is a complement for all the subspaces Im(df(z)), z ∈ U, then π is injective on all these subspaces. The map π ∘ f : U ⟼ E_{1} is a sub-mersion at z_{0}, because its differential is surjective and Ker(d(π ∘ f(z_{0})) = Ker(df(z_{0})), which is complemented in E. By Proposition 2.1, it follows that there exists an open neighborhood Z of z_{0} in U and a C^{k}-diffeomorphism α : V_{1} × V_{2} ⟼ Z, where V_{i} is an open neighborhood of 0 in E_{i}, i = 1, 2, such that α(0, 0) = z_{0} and the map π ∘ f ∘ α : V_{1} × V_{2} ⟼ E_{1} is given by π(f(α(x_{1}, x_{2}))) = x_{1}.

Hence, in V_{1} × V_{2}, the map f ∘ α : V_{1} × V_{2} ⟼ E_{1} ⊕ F_{1} is given by:

f(α(x_{1}, x_{2})) = (x_{1}, λ(x_{1}, x_{2})),

where λ : V_{1} × V_{2} ⟼ F_{1} is a map of class C^{k}.

We claim that λ does not depend on the variable x_{2}. To show this, we assume that V_{2} is convex (otherwise, we pass to a smaller convex subset of V_{2} containing 0) and we show that the derivative ∂_{2}λ of λ with respect to the second variable is identically zero. To this aim, for all (x_{1}, x_{2}) ∈ V_{1} × V_{2}, let’s denote by E_{x_{1},x_{2}} the image of d(f ∘ α)(x_{1}, x_{2})[ .] in F. Since α is a diffeomorphism, E_{x_{1},x_{2}} = Im(df(α(x_{1}, x_{2}))); in particular, E_{0,0} = E_{1} and π is the identity on E_{0,0}.

By the choice of π, it follows that the restriction π : E_{x_{1},x_{2}} ⟼ E_{1} to each of the spaces E_{x_{1},x_{2}} is an isomorphism.

If ∂_{2}λ(x_{1}, x_{2})[v_{2}] ≠ 0 for some (x_{1}, x_{2}) ∈ V_{1} × V_{2} and some v_{2} ∈ E_{2} \ {0}, then the vector

v = d(f ∘ α)(x_{1}, x_{2})[(0, v_{2})] = (0, ∂_{2}λ(x_{1}, x_{2})[v_{2}]) ∈ E_{x_{1},x_{2}}

would be a non zero vector such that π(v) = 0. This contradicts the injectivity of π on E_{x_{1},x_{2}} and proves that λ does not depend on the variable x_{2}.

Let’s consider the injection ι : V_{1} ⟼ V_{1} × V_{2} given by ι(x) = (x, 0); the composition f ∘ α ∘ ι is given by:

f(α(ι(x)) = (x, λ(x, 0)),

and its derivative is computed easily as:

d(f ∘ α ∘ ι)(x)[v] = (v, ∂_{1}λ(x, 0)[v]), ∀ x ∈ V_{1}, v ∈ E_{1}.

It follows that d(f ∘ α ∘ ι)(x) is injective for all x ∈ V_{1}.

Moreover, Im(d(f ∘α∘ι)(0)) = Im(df(z_{0})), which is complemented in F, and so f ∘ α ∘ ι is an immersion at x = 0.

By Proposition 2.1, there exists an open neighborhood V′_{1} ⊆ V_{1} of 0 in E_{1}, an open neighborhood V′_{2} of 0 in F_{1}, an open neighborhood W of 0 in F and a C^{k}-diffeomorphism β : W ⟼ V′_{1} × V′_{2} such that β(0) = (0, 0) and:

β(f(α(ι(x))) = (x, 0), ∀ x ∈ V′_{1}.  (2.3.1)

Since λ does not depend on the second variable, it follows:

f(α(ι(x_{1}))) = f(α(x_{1}, x_{2})), ∀ (x_{1}, x_{2}) ∈ V′_{1} × V′_{2}.  (2.3.2)

Finally, from (2.3.1) and (2.3.2) it follows:

β(f(α(x_{1}, x_{2}))) = (x_{1}, 0), ∀ (x_{1}, x_{2}) ∈ V′_{1} × V′_{2},

which concludes the proof.

We use the following strong notion of submanifold (see [5]): Definition 2.4. Let X be a Banach manifold of class C^{k}, k ≥ 1, modeled on a Banach space E. A subset X_{1} of X is a submanifold of X of class C^{k}, if for every x ∈ X_{1} there exists a neighborhood U_{x} of x in X and a chart ϕ_{x} : U_{x} ⟼ V_{x} of class C^{k}, where V_{x} is an open set of E, and a closed complemented subspace S of E such that:

ϕ_{x}(U_{x} ∩ X_{1}) = V_{x} ∩ S.

We call such a pair (U_{x}, ϕ_{x}) a submanifold chart for X_{1}.

We have the following corollary to Proposition 2.3:

Corollary 2.5. Let X and Y be two Banach manifolds of class C^{k}, k ≥ 1, modeled respectively on the Banach spaces E and F. Let f : X ⟼ Y be a map of class C^{k}. Let y_{0} ∈ f(X) ⊂ Y and X_{1} = f^{−1}(y_{0}). If f has locally constant rank on X_{1}, then X_{1} is a C^{k}-submanifold of X. For all z ∈ X_{1}, the tangent space T_{z}X_{1} is given by Ker(df(z)).

Proof: Given the local nature of the problem, it suffices to consider the case where X = U is an open subset of E and Y = F, and we can assume that f has constant rank in U.

Let z_{0} be a fixed point in X_{1}, E_{1} = Im(df(z_{0})) ⊂ F and E_{2} = Ker(df(z_{0})). By Proposition 2.3, given any z ∈ X_{1} there exists an open neighborhood Z of z in U which is C^{k}-diffeomorphic to the product V′_{1} × V′_{2}, with V′_{i} open subset of E_{i}, i = 1, 2. The set X_{1} ∩ V is identified, through this diffeomorphism, with the set {0}×V′_{2}, which is a submanifold of V′_{1} ×V′_{2}. Hence, X_{1} is a submanifold of E, modeled on the space E_{1}.

In the notation of Proposition 2.3, if π : E_{1}⊕F_{1} ⟼ E_{1} denotes the projection onto the first summand, then X_{1} ∩ Z = (π ∘ β ∘ f)^{−1}(0), and the map π ∘ β ∘ f is a sub-mersion in X_{1} ∩ Z. Hence, T_{z}X_{1} = Ker(dπ(dβ(df(z))), and since dβ is an isomorphism and Im(dβ ∘ df) ∩ Ker(dπ) = {0}, then T_{z}X_{1} = Ker(df(z)).

We present two more Corollaries to Proposition 2.3 that concern the manifold structure for the image of a map with locally constant rank.

Corollary 2.6. Let X and Y be Banach manifolds of class C^{k}, k ≥ 1, and f : X ⟼ Y be a C^{k} map having locally constant rank. If f : X ⟼ f(X) is an open map (with respect to the induced topology in f(X)), then f(X) is a C^{k}-submanifold of Y, and for every x ∈ X, T_{f(x)}f(X) = Im(df(x)).

Proof: The Rank Theorem gives us, for every x ∈ X, coordinate systems ϕ : U ⟼ Ũ in X and ψ : V ⟼ Ṽ in Y such that x ∈ U, f(U) ⊂ V, Ũ is open in E_{1} ⊕ E_{2}, and Ṽ open in E_{1} ⊕ F such that f̃ = ψ ∘ f ∘ ϕ^{−1} is given by f̃(x, y) = (x, 0). Since f(U) ⊂ f(X) is open, we can find an open set Z ⊂ Y such that f(U) = Z ∩ f(X).

Let W = ψ^{−1}((Ũ × F) ∩ Ṽ) ∩ Z and W̃ = ψ(W). Then, ψ(f(W) ∩ W) = (E_{1} ⊕ 0) ∩ W̃, i.e., ψ|_{W} is a submanifold chart for f(X) around f(x).

For idempotent maps we can prove the same result of Corollary 2.6 without using the topological condition on f(X):

Corollary 2.7. Let X be a Banach manifold of class C^{k}, k ≥ 1, and f : X ⟼ X be a C^{k}-map having locally constant rank, and such that f ∘ f = f (i.e., f fixes the points of f(X)). Then, f(X) ⊂ X is a C^{k}-submanifold of X, and, for all x ∈ f(X), T_{x}f(X) = Im(df(x)).

Proof: Let x ∈ f(X) be fixed. The Rank Theorem gives us charts ϕ : U ⟼ Ũ and ψ : V ⟼ Ṽ around x in X such that f(U) ⊂ V and f̃ = ψ ∘ f ∘ ϕ^{−1} : Ũ ⟼ Ṽ is given by f̃(x, y) = (x, 0), where Ũ is open in E_{1} ⊕ E_{2} and Ṽ is open in E_{1} ⊕ F.

Let U′ = U ∩ f^{−1}(U). Then U′ ⊂ U is open, x ∈ U′ and f(U′) ⊂ U′.

Let Ũ′ = ϕ(U′), V′ = ψ^{−1}(Ũ′ × F) ∩ Ṽ) ∩ U′ and Ṽ′ = ψ(V′). Then, f(U′) = f(X) ∩ V′ and ψ|_{V′} is a submanifold chart for f(X) around x, i.e., ψ(f(X) ∩ V′) = (E_{1} ⊕ 0) ∩ Ṽ′.

3. The Differentiable Structure for C_{b}(M,N)

In this section we consider a topological space M and a differentiable manifold N of class C^{k}, with k ≥ 3. The manifold N is assumed to be Hausdorff and second countable, while no specific assumption is made on the topological space M.

As an example for our theory, we consider C_{b}(M, N) the set of all continuous maps from M to N whose image is precompact in N:

C_{b}(M, N) = \{f ∈ C(M, N) : \overline{f(M)} is compact in N\}.

Observe that C_{b}(M, IR^{n}) is a Banach space with respect to the norm:

∥f∥_{∞} = sup_{x∈M} |f(x)|.

Using Whitney’s Theorem, we assume for the moment that N is embedded as a C^{k}-submanifold of IR^{n}. Since N has the relative topology, we have:

C_{b}(M, N) = \{f ∈ C_{b}(M, IR^{n}) : \overline{f(M)} ⊆ N\}.

Given any subset U of IR^{n}, we denote by C_{b}(M, U) the subset of C_{b}(M, IR^{n}) consisting of those function f such that the closure \overline{f(M)} is contained in U.

It is an easy observation that, if U is an open set, then C_{b}(M, U) is an open subset of the Banach space C_{b}(M, IR^{n}). Namely, for f ∈ C_{b}(M, U), if we set ε = dist(\overline{f(M)}, IR^{n} \ U) > 0, then the open ball B(f, \frac{ε}{2}) in C_{b}(M, IR^{n}) centered at f and of radius \frac{ε}{2} is contained in C_{b}(M, U). Thus, C_{b}(M, U) is a smooth submanifold of the Banach space C_{b}(M, IR^{n}).

Let U be a tubular open neighborhood of N in IR^{n} and r : U ⟼ N a retraction of class C^{k−1}. Such a map r is built by restricting the exponential map of IR^{n} to the normal bundle T N^{⊥} of N in IR^{n}. This map gives a C^{k−1}- diffeomorphism between an open neighborhood of the null section of T N^{⊥} and an open neighborhood U of N in IR^{n} (see [5], Ch. IV, §5-6).

More explicitly, the map r is uniquely determined by the following property:

(r) there exists a continuous function ε_{0} : N ⟼ IR^{+} such that, for all x ∈ U there exists a unique p ∈ N such that, setting v = x − p, it is:

{(a) |v| < ε_{0}(p); \atop (b) v ∈ T_{p}N^{⊥}.}

Then, it is r(x) = p. It is easy to see that r is the identity on N, r(U) = N and so it is an

idempotent:

r ∘ r = r.  (3.0.1)

By shrinking U, if necessary, we can assume that r : U ⟼ N is a submersion, and therefore r : U ⟼ IR^{n} has constant rank.

We prove now a differentiability property for the composition map in the spaces C_{b}(M, ·).

Proposition 3.1. Let U be an open subset of IR^{n} and T : U ⟼ IR^{m} be a map of class C^{l}, with l ≥ 1. Then, the composition map

(T ∘) : C_{b}(M, U) ⟼ C_{b}(M, IR^{m})

given by:

(T ∘)(f) = T ∘ f

is a map of class C^{l−1}.

Proof: First of all we observe that (T ∘) is well defined. Namely, if f is in C_{b}(M, U) then T ∘ f is continuous; moreover, (T ∘ f)(M) ⊂ T (\overline{f(M)}), which is compact, and so T ∘ f ∈ C_{b}(M, IR^{m}).

Now, we show that if T is of class C^{1} then (T ∘) is continuous. To see this, we fix f ∈ C_{b}(M, U) and we denote by ε the distance (positive) between \overline{f(M)} and IR^{n} \ U.

Then, if g ∈ C_{b}(M, U) satisfies ∥g − f∥_{∞} < \frac{ε}{2}, then the segment I_{f(x),g(x)} with endpoints f(x) and g(x) is entirely contained in U for all x ∈ M.

Therefore, we have:

∥T ∘ g − T ∘ f∥_{∞} = sup_{x∈M} |T (g(x)) − T (f(x))| ≤

≤ sup_{x∈M} [ sup_{y∈I_{f(x),g(x)}} ∥dT (y)∥] · |f(x) − g(x)|.

(3.1.1)

Let K be the compact set:

K = \{y ∈ IR^{n} : dist(y, f(M)) ≤ \frac{ε}{2}\};

if we set

C = max_{y∈K} ∥dT (y)∥,  (3.1.2)

then, from (3.1.1) and (3.2.2) we obtain

∥T ∘ g − T ∘ f∥_{∞} ≤ C · ∥g − f∥_{∞}.  (3.1.3)

The inequality (3.1.3) implies that (T ∘) is locally Lipschitz.

To prove the differentiability of (T ∘), we use the following criterion for the differentiability of maps between Banach spaces. We recall that, given a Banach space F, if Λ is a family of linear maps from F to some other Banach space, Λ separates the points of F if for all x, y ∈ F, with x ≠ y, there exists λ ∈ Λ such that λ(x) ≠ λ(y). Equivalently, Λ separates the points of F if for all x ≠ 0 in F there exists λ ∈ Λ such that λ(x) ≠ 0.

Proposition 3.2. Let E and F be real Banach spaces, E′ be an open subset of E, f : E′ ⟼ F a map, and g : E′ ⟼ L(E, F) a continuous map. Suppose that there exists a family Λ of continuous linear maps from F to IR^{s} that separates the points of F and such that, for all h ∈ E′, v ∈ E and for all λ ∈ Λ, it is:

\frac{d}{dt}|_{t=0}λ(f(h + tv)) = λ(g(h)v).

Then, f is a map of class C^{1} on E′ and df = g.

Proof: Let h ∈ E′ be fixed. For all w ∈ E with ∥w∥ small enough, we set R(w) = f(h+w)−f(h)−g(h)w; since g(h) is linear and continuous, to prove that f is differentiable at h and df(h) = g(h) it suffices to show that lim_{w→0} R(w) · ∥w∥^{−1} = 0.

To this aim, consider the curve γ(t) = g(h+tw)w in F. Since γ is continuous, then there exists the F -valued integral:

I = ∫_{0}^{1} γ(t) dt ∈ F.

We claim that I = f(h + w) − f(h). Indeed, if it were I ≠ f(h + w) − f(h), by the separability condition on Λ there would exist λ ∈ Λ such that

λ(I) ≠ λ(f(h + w)) − λ(f(h)).  (3.2.1)

By the continuity of λ, it is:

λ(I) = λ (∫_{0}^{1} γ(t) dt) = ∫_{0}^{1} λ(γ(t)) dt = ∫_{0}^{1} λ(g(h + tw)w) dt. (3.2.2)

On the other hand, our differentiability condition on the function

q(t) = λ(f(h + tw))

implies that q′(t) = λ(g(h + tw)w), hence, from (3.2.2) we get:

\begin{aligned}λ(f(h + w)) − λ(f(h)) & = q(1) − q(0) = ∫_{0}^{1} q′(t) dt = \\ & = ∫_{0}^{1} λ(g(h + tw)w) dt = λ(I). \end{aligned}  (3.2.3)

The identity (3.2.3) contradicts (3.2.1), hence

R(w) = ∫_{0}^{1} (g(h + tw)w − g(h)w) dt.

Since the evaluation at w is a continuous function in L(E, F), then we have:

R(w) = (∫_{0}^{1} (g(h + tw) − g(h)) dt) w,

from which we get:

∥R(w)∥ ≤ ∥w∥ ∫_{0}^{1} ∥g(h + tw) − g(h)∥ dt.

By the continuity of g, the latter integral tends to 0 as w → 0, so that ∥R(w)∥ · ∥w∥^{−1} goes to 0 as w → 0. This implies that df(h) = g(h). Since g is continuous, then f is C^{1} and we are done.

Going back to the proof of Proposition 3.1, we now assume that T is of class C^{2} and we prove that (T ∘) is a map of class C^{1}.

We introduce the following notation. Let O denote the map:

O : C_{b}(M, L(IR^{n}, IR^{m})) ⟼ L (C_{b}(M, IR^{n}), C_{b}(M, IR^{m}))

given by:

O(A)(f)(x) = A(f(x)),

where A ∈ C_{b}(M, L(IR^{n}, IR^{m})), f ∈ C_{b}(M, IR^{n}) and x ∈ M.

The map O is well defined, linear and continuous. To see this, let B denote the map:

\begin{matrix}B : L(IR^{n}, IR^{m}) & ⊕ IR^{n} ⟼ IR^{m} \\ (a, & x) ⟼ a(x), \end{matrix}

which is clearly bilinear and C^{1}. By the first part of the proof, this implies that the map:

(B∘) : C_{b}(M, L(IR^{n}, IR^{m}) ⊕ IR^{n}) ⟼ C_{b}(M, IR^{n})

is continuous. Thus, we can write O(A) as:

O(A)(f) = (B∘)(A ⊕ f),

for all A ∈ C_{b}(M, L(IR^{n}, IR^{m})) and f ∈ C_{b}(M, IR^{n}), which implies that O is well defined and continuous. The linearity of O is trivial.

Finally, we are ready to prove that (T ∘) is C^{1} by using Proposition 3.2, and to show that:

d(T ∘) = O ∘ (dT ∘).  (3.2.4)

To this aim, we consider the Banach spaces E = F = C_{b}(M, IR^{n}), the open subset of E given by C_{b}(M, U), and the family Λ = {λ_{x}}_{x∈M} of linear functionals from F to IR^{n} given by the evaluations maps:

λ_{x}(h) = h(x), h ∈ C_{b}(M, IR^{n}).

Clearly, Λ separates the points of F. Moreover, in the notation of Proposition 3.2 we set f = (T ∘) and g = O ∘ (dT ∘). By the first part of the proof, the map

g : C_{b}(M, U) ⟼ L(C_{b}(M, IR^{n}), C_{b}(M, IR^{n}))

is continuous. Moreover, for h ∈ C_{b}(M, U), v ∈ C_{b}(M, IR^{n}) and x ∈ M, it is:

\begin{aligned}\frac{d}{dt}|_{t=0}λ_{x}((T ∘)(h + tv)) & = \frac{d}{dt}|_{t=0}(T (h(x) + tv(x)) = dT (h(x))[v(x)] \\ & = λ_{x}(dT (h(·))[v(·)] = λ_{x}(O ∘ (dT ∘)(h)[v]). \end{aligned}

So, by Proposition 3.2, the map (T ∘) is differentiable, and its differential is given by O ∘ (dT ∘). With a little abuse of notation, we will write:

d(T ∘) = (dT ∘).

Finally, an easy induction argument shows that, if T is of class C^{l}, then (T ∘) is of class C^{l−1}.

Lemma 3.3. Let T : U ⊂ IR^{m} ⟼ IR^{n} be a C^{1}-map with constant rank and f ∈ C_{b}(M, U). Then there exists ε > 0 such that, for all q ∈ \overline{f(M)} and p ∈ U with |q − p| < ε, then

Im(dT (q))^{⊥} ⊕ Im(dT (p)) = IR^{n}.

Proof: Let p ∈ U be fixed. There exists an open neighborhood U_{p} of p in U such that, for every q_{1}, q_{2} ∈ U_{p} it is:

Im(dT (q_{1})) ⊕ Im(dT (q_{2})) = IR^{n}.  (3.3.1)

Namely, since T has constant rank, to prove (3.3.1) it suffices to show that Im(dT (q_{1})) + Im(dT (q_{2})) = IR^{n}. Such a condition is open on q_{1} and q_{2}, thus it holds in an open neighborhood of p.

The closure \overline{f(M)} is compact in U; let ε′ = dist(\overline{f(M)}, IR^{m} \ U). The set K = \{p ∈ IR^{m} : dist(p, \overline{f(M)}) ≤ \frac{ε′}{2} \} ⊂ U is compact.

By the Lebesgue’s Covering Lemma there exists ε > 0, ε ≤ \frac{ε′}{2} , such that every subset of K with diameter less than ε is contained in some U_{p}.

Then, if q ∈ \overline{f(M)}, p ∈ U and |q − p| < ε it follows that q, p ∈ K and so q, p ∈ U_{r} for some r ∈ K, and we are done.

The following result will be proven in a more general context in the next section. Some of the details of the proof will be left to the proof of Theorem 4.2.

Proposition 3.4. Let T : U ⊂ IR^{m} ⟼ IR^{n} be a C^{2}-map having constant rank. Then, the map (T ∘) : C_{b}(M, U) ⟼ C_{b}(M, IR^{n}) has locally constant rank.

Proof: Let f ∈ C_{b}(M, U) be fixed; we need to show that (T ∘) has constant rank in a neighborhood of f.

Clearly, from (3.2.4) we obtain easily:

Ker(d(T ∘)(f)) = {v ∈ C_{b}(M, IR^{n}) : v(x) ∈ Ker(dT (f(x))}.  (3.4.1)

Moreover, it is:

Im(d(T ∘)(f)) = {v ∈ C_{b}(M, IR^{n}) : v(x) ∈ Im(dT (f(x)))}.  (3.4.2)

Namely, it is not hard to prove that, for all v ∈ C_{b}(M, IR^{n}) it is possible to determine a continuous and bounded map w : M ⟼ IR^{n} such that d(T ∘)(f)[w] = v. For instance, one can take the unique w satisfying w(x) ∈ ker(dT (f(x)))^{⊥} and such that dT (f(x))[w(x)] = v(x), for all x ∈ M.

From (3.4.1) and (3.4.3) it follows immediately that the spaces Ker(d(T ∘)(f)) and Im(d(T ∘)(f)) are closed. It is easy to see that the space

{v ∈ C_{b}(M, IR^{n}) : v(x) ∈ Ker(dT (f(x)))^{⊥}}  (3.4.3)

is a closed complement for Ker(d(T ∘)(f)).

To prove that the spaces Im(d(T ∘)(g)) have a common complement for g ∈ C_{b}(M, U) in a neighborhood of f, we take ε > 0 as in Lemma 3.3.

Then, if ∥g − f∥_{∞} < ε, we have:

Im(dT (f(x)))^{⊥} ⊕ Im(dT (g(x))) = IR^{n}, ∀ x ∈ M.

From this, it follows easily that

{v ∈ C_{b}(M, IR^{n}) : v(x) ∈ Im(dT (f(x)))^{⊥}}  (3.4.4)

is a closed complement for all Im(d(T ∘)(g)), with g in a ball of radius less than ε around f.

Theorem 3.5. C_{b}(M, N) is a submanifold of class C^{k−2} of the Banach space C_{b}(M, IR^{n}). For all f ∈ C_{b}(M, N), the tangent space T_{f}C_{b}(M, IR^{n}) is given by:

T_{f}C_{b}(M, N) = {v ∈ C_{b}(M, IR^{n}) : v(x) ∈ T_{f(x)}N, }.

Proof: Since r has constant rank, it follows from Proposition 3.4 that the map (r∘) : C_{b}(M, U) ⟼ C_{b}(M, U) has locally constant rank. Then, by Corollary 2.7, it follows that the image of (r∘) is a C^{k−2}-submanifold of C_{b}(M, IR^{n}).

Since N is the set of fixed points of r, it follows that N is closed in U, and therefore the image of (r∘) is precisely C_{b}(M, N).

4. Two Axioms for the General Case.

In this section we generalize the construction made in the previous section to a general set of maps 𝔐(M, N) satisfying some axioms.

Let M be a topological space. Suppose we are given a Banachable space 𝔐(M, IR) of functions f : M ⟼ IR (i.e., 𝔐(M, IR) is a topological space whose topology is compatible with a norm which makes 𝔐(M, IR) into a Banach space).

Under certain assumptions for 𝔐(M, IR) we wish to build a functor 𝔐 from the category of differentiable finite dimensional manifolds (of some regularity C^{k}) to the category of Banach differentiable manifolds (of some regularity C^{m}). More precisely, for every finite dimensional manifold N we shall associate a set of maps 𝔐(M, N) with a Banach manifold structure and for every map T : N_{1} ⟼ N_{2} we shall associate a map 𝔐(T) = (T ∘) : 𝔐(M, N_{1}) ⟼ 𝔐(M, N_{2}) given by 𝔐(T)(f) = T ∘ f.

First let 𝔐(M, IR^{n}) = ⨁_{k=1}^{n} 𝔐(M, R), i.e., 𝔐(M, IR^{n}) is the topological space of maps f : M ⟼ IR^{n} whose coordinates f_{i} are in 𝔐(M, IR) and the topology of 𝔐(M, IR^{n}) is the product topology of n copies of 𝔐(M, IR).

If S ⊂ IR^{n} is any subset, we denote by 𝔐(M, S) the set of maps f ∈ 𝔐(M, IR^{n}) such that \overline{f(M)} ⊂ S. Consider 𝔐(M, S) with the induced topology of 𝔐(M, IR^{n}). We axiomatize the following two properties: (Ax1) 𝔐(M, IR) ⊂ C_{b}(M, IR) and the inclusion is continuous. (Ax2) If U is an open subset of IR^{n} and T : U ⟼ IR^{m} is a map of

class C^{l}, then the map (T ∘) : 𝔐(M, U) → 𝔐(M, IR^{m}) given by (T ∘)(f) = T ∘ f is (well-defined and) continuous.

The integer l ≥ 0 in (Ax2) will be fixed throughout the rest of the section. For instance, we proved in the previous section that 𝔐 = C_{b} satisfies the axioms (Ax1) and (Ax2) with l = 1.

It follows easily from (Ax1) that 𝔐(M, IR^{n}) ⊂ C_{b}(M, IR^{n}) with continuous inclusion. This implies that the domain 𝔐(M, U) of (T ∘) in (Ax2) is open.

Observe also that, in view of Axiom (Ax2), the definition of 𝔐(M, IR^{n}) as the sum ⨁_{k=1}^{n} 𝔐(M, IR) is the only possibility. Indeed, if π_{α} : IR^{n} ⟼ IR is the projection on the α-th coordinate and i_{α} : IR ⟼ IR^{n} is the inclusion in the α-th coordinate, then the maps:

((π_{1}∘), (π_{2}∘), …, (π_{n}∘))

and

(i_{1}∘) + (i_{2}∘) + … + (i_{n}∘)

are mutually inverse topological isomorphisms between the spaces 𝔐(M, IR^{n}) and ⨁_{k=1}^{n} 𝔐(M, IR).

We first generalize Proposition 3.1:

Proposition 4.1. Let M be a topological space, and let 𝔐(M, IR) be a Banachable space satisfying Axioms (Ax1) and (Ax2). Let U be an open subset of IR^{n} and T : U ⟼ IR^{m} be a map of class C^{k}, with k ≥ l. Then, the composition map (T ∘) : 𝔐(M, U) ⟼ 𝔐(M, N) is of class C^{k−l}.

Proof: The proof of Proposition 3.1 can be repeated verbatim, except for the fact that now the first part of the proof is simply the axiom (Ax2).

Theorem 4.2. Let M be a topological space, and let 𝔐(M, IR) be a Banachable space satisfying Axioms (Ax1) and (Ax2). Let N be a C^{l+k} submanifold of IR^{n}, with k ≥ 2. Then, the set 𝔐(M, N) is a submanifold of class C^{k−1} of the Banach space 𝔐(M, IR^{n}), and for all f ∈ 𝔐(M, N) the tangent space T_{f}𝔐(M, N) is given by:

T_{f}𝔐(M, N) = {v ∈ 𝔐(M, IR^{n}) : v(x) ∈ T_{f(x)}N}.

Proof: We proceed in the same way as in the proof of Theorem 3.5. We only need to generalize the results of Proposition 3.4.

We assume here that T is of class C^{l+1}. Let’s first verify the generalization of (3.4.2). Let f ∈ 𝔐(M, U) and let w ∈ 𝔐(M, IR^{n}) be such that w(x) ∈ Im(dT (f(x))) for all x ∈ M. Define v : M ⟼ IR^{m} by requiring that v(x) ∈ Ker(dT (f(x)))^{⊥}, and dT (f(x))[v(x)] = w(x) for all x ∈ M.

We need to show that v ∈ 𝔐(M, IR^{m}); it will follow that w = d(T ∘)(f)[v] and thus w ∈ Im(d(T ∘)(f)). To this aim, define A : U × IR^{n} ⟼ IR^{m} in such a way that, if p ∈ U and w_{0} ∈ IR^{n}, then A(p, w_{0}) ∈ IR^{m} is the unique element v_{0} ∈ Ker(dT (p))^{⊥} for which dT (p)[v_{0}] is the orthogonal projection of w_{0} onto Im(dT (p)).

Using orthonormal moving frames in IR^{n} and IR^{m} adapted to the distributions (of class C^{l}) given by Ker(dT (p)) and Im(dT (p)), one checks easily that A is a map of class C^{l}. Now, v = (A∘)(f, w), and, since (f, w) ∈ 𝔐(M, U × IR^{n}), it follows from Axiom (Ax2) that v ∈ 𝔐(M, IR^{m}).

Let’s now show that the spaces Ker(d(T ∘)(f)) and

Z_{f} = {v ∈ 𝔐(M, IR^{n}) : v(x) ∈ Ker(dT (f(x)))^{⊥}}

are closed and complementary to each other. Observe that, since the convergence in the space 𝔐(M, IR^{m}) implies uniform convergence, and in particular pointwise convergence, it is clear that the two spaces above are closed.

It is also obvious that Ker(d(T ∘)(f)) ∩ Z_{f} = {0}.

If v ∈ 𝔐(M, IR^{n}), we can write v uniquely as the sum v_{1} + v_{2}, with v_{1}(x) ∈ Ker(dT (f(x)) and v_{2} ∈ Ker(dT (f(x)))^{⊥} for all x ∈ M. We want to show that v_{1} and v_{2} are in 𝔐(M, IR^{n}).

Define P : U ×IR^{n} ⟼ IR^{n} by taking P (p, v_{0}) to be the orthogonal projection of v_{0} onto Ker(dT (p)) for all p ∈ U and v_{0} ∈ IR^{n}. Then, P is of class C^{l} and thus, by Axiom (Ax2), v_{1} = (P ∘)(f, v) and v_{2} = v − v_{1} are in 𝔐(M, IR^{n}).

Finally, to conclude the proof, we need to show that, using the same notations of Proposition 3.4, for g ∈ 𝔐(M, U) such that ∥g − f∥_{∞} < ε, the spaces Im(d(T ∘)(g)) and

W_{f} = {v ∈ 𝔐(M, IR^{n}) : v(x) ∈ Im(dT (f(x)))^{⊥}}

are closed and complementary to each other.

Again, it is obvious that these spaces are closed and their intersection is trivial.

Let v ∈ 𝔐(M, IR^{n}); write v = v_{1} + v_{2}, where v_{1}(x) ∈ Im(dT (g(x))) and v_{2}(x) ∈ Im(dT (f(x)))^{⊥} for all x ∈ M. We need to show that v_{1} and v_{2} are in 𝔐(M, IR^{n}). Let V be the set of pairs (q, p) ∈ U×U such that Im(dT (p))⊕Im(dT (q))^{⊥} = IR^{n}. Define Q : V ×IR^{n} ⟼ IR^{n} by requiring that Q(q, p, v) be the first projection of v relative to the decomposition IR^{n} = Im(dT (p)) ⊕ Im(dT (q))^{⊥}. Using moving frames, it is not difficult to see that V is open and Q is of class C^{l}.

Observe that (g, f, v) ∈ 𝔐(M, V × IR^{n}) and v_{1} = (Q∘)(g, f, v), hence the fact that v_{1} ∈ 𝔐(M, IR^{n}) follows from the axiom (Ax2). Also, v_{2} = v − v_{1} ∈ 𝔐(M, IR^{n}), and we are done.

Let now N be a C^{k+l} (k ≥ 2) abstract manifold (i.e. not an embedded submanifold of IR^{n}). By Whitney’s Theorem we can find an embedding ϕ : N ⟼ IR^{n} of class C^{k+l}, for n large enough.

Then, ϕ(N) is a C^{k+l}-submanifold of IR^{n}, and the previous theorem says that, if the axioms (Ax1) and (Ax2) are satisfied, then 𝔐(M, ϕ(N)) is a Banach submanifold of 𝔐(M, IR^{n}) of class C^{k−1}. Let 𝔐(M, N, ϕ) be the set of maps f : M ⟼ N such that ϕ∘f ∈ 𝔐(M, ϕ(N)). The function

(ϕ∘) : 𝔐(M, N, ϕ) ⟼ 𝔐(M, ϕ(N))

is a bijection, and therefore it induces a unique C^{k−1} Banach manifold structure in 𝔐(M, N, ϕ).

The notation 𝔐(M, N, ϕ) is temporary, as we will prove that 𝔐(M, N, ϕ) does not depend on the choice of the embedding ϕ, and therefore we will simply write 𝔐(M, N).

The next theorem shows the functoriality of 𝔐.

Theorem 4.3. Let M be a topological space and let 𝔐(M, IR) be a Banachable space satisfying the axioms (Ax1) and (Ax2). Let N_{1} and N_{2} be C^{k+l} manifolds (k ≥ 2) and let ϕ_{i} : N_{i} ⟼ IR^{n_{i}}, i = 1, 2, be C^{k+l}-embeddings.

Let T : N_{1} ⟼ N_{2} be a C^{k+l}-map. Then, the map (T ∘) : 𝔐(M, N_{1}, ϕ_{1}) ⟼ 𝔐(M, N_{2}, ϕ_{2}) is well defined and of class C^{k−1}.

Proof: Denote by T′ : ϕ_{1}(N_{1}) ⟼ ϕ_{2}(N_{2}) be the map T′ = ϕ_{2}∘T ∘ϕ_{1}^{−1}. Clearly, T′ is of class C^{k+l}.

By definition of the differentiable structure of 𝔐(M, N_{i}, ϕ_{i}), the maps

(ϕ_{i}∘) : 𝔐(M, N_{i}, ϕ_{i}) ⟼ 𝔐(M, ϕ_{i}(N_{i}))

are diffeomorphisms (i = 1, 2), and so we only need to prove that (T′∘) : 𝔐(M, ϕ_{1}(N_{1})) ⟼ 𝔐(M, ϕ_{2}(N_{2})) is of class C^{k−1}.

To show this we argue as follows. As in the previous section, we can find a map r : U ⟼ ϕ_{1}(N_{1}) of class C^{k+l−1}, defined in an open set U ⊂ IR^{n_{1}} containing ϕ_{1}(N_{1}), such that r|_{ϕ_{1}(N_{1})} is the identity map. Then, T′ ∘ r : U ⟼ IR^{n_{2}} is an extension of T′ of class C^{k+l−1}.

It follows from Proposition 4.1 that (T′ ∘ r∘) is of class C^{k−1}, and (T′∘) is just the restriction of (T′ ∘ r∘), which concludes the proof.

Finally, we prove that the space 𝔐(M, N, ϕ) and its differential structure do not depend on the embedding ϕ:

Corollary 4.4. Under the same hypotheses of Theorem 4.3, 𝔐(M, N, ϕ_{1}) and 𝔐(M, N, ϕ_{2}) coincide, and they carry the same C^{k−1}-differentiable structure.

Proof: It suffices to take T = Id (the identity map on N) in Theorem 4.3. The fact that (T ∘) : 𝔐(M, N, ϕ_{1}) ⟼ 𝔐(M, N, ϕ_{2}) is well defined shows that

𝔐(M, N, ϕ_{1}) ⊆ 𝔐(M, N, ϕ_{2})

and that the inclusion map is of class C^{k−1}. By interchanging the role of ϕ_{1} and ϕ_{2} we obtain the opposite inclusion and the thesis.

5. The Case of Functions Vanishing at Infinity

In this section we discuss a family of examples of Banach spaces of real functions on a topological space M that were left out in the axiomatization of the previous section. Namely, we consider now spaces of functions contained in the space C_{0}(M, IR), the set of continuous functions vanishing at infinity, with the supremum norm. Observe indeed that, in this case, the axiom (Ax2) is not satisfied by all functions T of class C^{1} (nor of class C^{l} with l ≥ 1), but only by those C^{1}-functions T satisfying T (0) = 0.

However, Proposition 3.1 can be easily generalized to the case of C_{0} if we add the hypotheses 0 ∈ U and T (0) = 0. To see this, we observe the following Lemma:

Lemma 5.1. Let E and F be two Banach spaces, E′ ⊂ E an open subset and S ⊂ F a closed subspace. Denote by i : S ⟼ F the inclusion map. Then, a map f : E′ ⟼ S is of class C^{k} (k ≥ 0) if and only if i ∘ f : E′ ⟼ F is of class C^{k}.

Proof: Observe that the result is trivial if S is complemented in F (for instance when S has finite codimension in F, or if F is a Hilbert space) because in this case there exists a continuous linear projection R : F ⟼ S, and f = R ∘(i∘f). In the general case the proof is done by induction on k, by looking explicitly at the definition of differentiability.

The case k = 0 follows from the fact that, being a closed subspace, S has the relative topology of F.

For the case k = 1, it suffices to observe that the differential d(i ∘ f)(x) takes values in S, since S is closed and d(i ∘ f)(x)[v] is just the directional derivative, thus it is limit of elements in S. Finally, for the induction step, we observe that the space L_{k}(E, S) of k-linear continuous maps from E to S can be identified with a closed subspace of L_{k}(E, F).

Now, to generalize Proposition 3.1 to the case of C_{0}, observe that C_{0}(M, IR^{n}) is a closed subspace of C_{b}(M, IR^{n}), and that, if T (0) = 0, then (T ∘) takes C_{0}(M, U) into C_{0}(M, IR^{n}).

Having observed this, it is now easy to generalize Proposition 3.5 to the case of C_{0}(M, N) if we add the hypothesis 0 ∈ N.

Motivated by this example, we now consider the following setup.

Let 𝔐_{0}(M, IR) be a Banachable space of real valued functions on M. Again, define 𝔐_{0}(M, IR^{n}) = ⨁_{k=1}^{n} 𝔐(M, IR) for n ≥ 1. If S ⊂ IR^{n} and 0 ∈ S, we define 𝔐_{0}(M, S) to be the set of maps f ∈ 𝔐_{0}(M, IR^{n}) such that \overline{f(M)} ⊂ S.

We axiomatize the following two properties:

(Ax1)′ 𝔐_{0}(M, IR) ⊂ C_{0}(M, IR) and the inclusion is continuous. (Ax2)′ If U is an open subset of IR^{n} containing 0 and T : U ⟼ IR^{m} is

a C^{l} map satisfying T (0) = 0, then the map (T ∘) : 𝔐_{0}(M, U) → 𝔐_{0}(M, IR^{m}) given by (T ∘)(f) = T ∘ f is (well-defined and) continuous.

We want to generalize Proposition 4.1 by using Lemma 5.1:

Proposition 5.2. Let M be a topological space and let 𝔐_{0}(M, IR) be a banachable space satisfying the axioms (Ax1)′ and (Ax2)′. Let U be an open set of IR^{n} containing 0 and T : U ⟼ IR^{m} be a map of class C^{k}, with k ≥ l, satisfying T (0) = 0. Then, the composition map (T ∘) : 𝔐_{0}(M, U) ⟼ 𝔐_{0}(M, IR^{m}) is of class C^{k−l}.

Proof: We proceed by induction on k. The case k = l is simply the axiom (Ax2)′. If k ≥ l + 1, we’ll prove that (T ∘) is of class C^{1} and that, in the notation of Proposition 3.1, d(T ∘) = O ∘ (dT ∘). Let 𝔐(M, IR^{p}) be the set of functions of the form f_{0}+c, where f_{0} ∈ 𝔐_{0}(M, IR^{p}) and c ∈ IR^{p}.

We make 𝔐(M, IR^{p}) into a Banachable space by declaring that the bijective map:

(f_{0}, c) ∈ 𝔐_{0}(M, IR^{p}) ⟼ f_{0} + c ∈ 𝔐(M, IR^{p})

be a topological isomorphism.

Notice that if f ∈ 𝔐(M, IR^{p}), then c = lim_{x→∞} f(x) exists, f_{0} = f − c ∈ 𝔐_{0}(M, IR^{p}) and f corresponds to the pair (f_{0}, c) by the above defined map.

It follows directly from axiom (Ax2)′ that the map

(dT ∘) : 𝔐_{0}(M, U) ⟼ 𝔐(M, L(IR^{n}, IR^{m}))

is (well defined and) continuous, since it can be identified with the map

((S∘), dT (0)),

where S = dT − dT (0).

Now, we need to prove that the linear map:

O : 𝔐(M, L(IR^{n}, IR^{m})) ⟼ L(𝔐_{0}(M, IR^{n}), 𝔐_{0}(M, IR^{m}))

defined as in (3.1.4) is (well defined and) continuous.

It will then follow from Proposition 3.2 (arguing as in the proof of Proposition 3.1) that d(T ∘) = O ∘ (dT ∘).

As we observed in the proof of Proposition 3.1, it suffices to show that the bilinear map:

(B∘) : 𝔐(M, L(IR^{n}, IR^{m})) × 𝔐_{0}(M, IR^{n}) ⟼ 𝔐_{0}(M, IR^{m})

is (well defined and) continuous, where B is defined as in (3.1.5).

To see this, we observe that the map (B∘) can be identified with the map:

𝔐_{0}(M, L(IR^{n}, IR^{m})) ⊕ L(IR^{n}, IR^{m}) ⊕ 𝔐_{0}(M, IR^{n}) ⟼ 𝔐_{0}(M, IR^{m})

given by:

(A, a, v) ⟼ (B∘)(A, v) + ∑_{i=1}^{m} ∑_{j=1}^{n} a_{ij}v_{j}e_{i},

where A = (a_{ij}), v = (v_{j}) and e_{1}, e_{2}, …, e_{m} is the canonical basis of IR^{m}.

The continuity of (A, v) ⟼ (B∘)(A, v) follows from the axiom (Ax2)′, and the continuity of the second term follows from the isomorphism 𝔐_{0}(M, IR^{n}) ≃ ⨁_{k=1}^{n} 𝔐_{0}(M, IR).

The induction step is now clear, and the proof is concluded.

Finally, we have the analogue of Theorem 4.2 for the case of 𝔐_{0}:

Theorem 5.3. Let M be a topological space and let 𝔐_{0}(M, IR) be a Banachable space satisfying the axioms (Ax1)′ and (Ax2)′. Let N be a C^{k+l} submanifold of IR^{n}, with k ≥ 2, with 0 ∈ N. Then, the set 𝔐_{0}(M, N) is a submanifold of class C^{k−1} of the Banach space 𝔐_{0}(M, IR^{n}), and, for all f ∈ 𝔐_{0}(M, N), the tangent space T_{f}𝔐_{0}(M, N) is given by:

T_{f}𝔐_{0}(M, N) = {v ∈ 𝔐_{0}(M, IR^{n}) : v(x) ∈ T_{f(x)}N}.

Proof: Just observe that in the proof of Theorem 4.2 all the applications of axioms (Ax2) can be replaced by the axiom (Ax2)′, since all the maps involved vanish at 0. For instance, since 0 ∈ N, the map r : U ⟼ N satisfies r(0) = 0.

If N is an abstract manifold of class C^{k+l}, with k ≥ 2, and ϕ : N ⟼ IR^{n} is a C^{k+l}-embedding, as in the previous section we can define 𝔐_{0}(M, N, ϕ), provided that 0 ∈ ϕ(N).

Theorem 4.3 can easily be generalized by adding the hypotheses T (p_{1}) = p_{2}, where p_{1} ∈ N_{1} and p_{2} ∈ N_{2} are such that ϕ_{1}(p_{1}) = ϕ_{2}(p_{2}) = 0. Also, Corollary 4.4 generalizes by adding the hypothesis that ϕ_{1} and ϕ_{2} vanish at the same point p on N. Thus, the differential structure of 𝔐_{0}(M, N, ϕ) depends only on the point p ∈ N such that ϕ(p) = 0. We then write 𝔐_{0}(M, N, p).

6. Examples

In this section we discuss some concrete examples of our theory.

6.1. The space C_{L}(M, N) of functions having limit at infinity

We denote by C_{L}(M, IR) the space of continuous functions f : M ⟼ IR for which there exists the limit lim_{x→∞} f(x) = c ∈ IR. Endowed with the supremum norm, C_{L}(M, IR) becomes a closed subspace of C_{b}(M, IR), and so C_{L}(M, IR) satisfies the axiom (Ax1).

If T : U ⊂ IR^{n} ⟼ IR^{m} is a map of class C^{1}, then, as we proved in Section 3, the map (T ∘) : C_{b}(M, U) ⟼ C_{b}(M, IR^{m}) is continuous. In particular, (T ∘) : C_{L}(M, U) ⟼ C_{b}(M, IR^{m}) is continuous, because C_{L}(M, IR) has the induced topology of C_{b}(M, U); moreover, it is clear that (T ∘)(C_{L}(M, U) ⊂ C_{L}(M, IR^{m}).

Again, since C_{L}(M, IR^{n}) is a closed subspace of C_{b}(M, IR^{m}), from Proposition 5.1 it follows that (T ∘) : C_{L}(M, U) ⟼ C_{L}(M, IR^{m}) is continuous, and so C_{L}(M, IR) satisfies the axiom (Ax2) with l = 1.

From Theorem 4.2 we have that, if N is a differentiable manifold of class C^{k+1}, with k ≥ 2, then the set C_{L}(M, N), which is the set of continuous functions f : M ⟼ N for which there exists the limit lim_{x→∞} f(x) ∈ N, has the structure

of a Banach manifold of class C^{k−1}.

6.2. The space C_{b}^{k}(M, N) of C^{k}-functions with bounded derivatives

If V_{1}, V_{2} are normed linear spaces, for i ≥ 1 we denote by L_{i}(V_{1}, V_{2}) the space of i-linear continuous maps from V_{1} to V_{2}; we set L_{0}(V_{1}, V_{2}) = V_{2}.

Let U ⊂ IR^{d} be an open set with C^{1}-boundary; for k ≥ 1 we denote by C_{b}^{k}(U, IR) the space of functions f : U ⟼ IR of class C^{k} such that the differentials d^{(i)}f : U ⟼ L_{i}(IR^{d}, IR) are bounded for all i = 0, 1, …, k.

Define a norm ∥ · ∥_{b}^{k} on C_{b}^{k}(U, IR) by:

∥f∥_{b}^{k} = ∑_{i=0}^{k} sup_{x∈U} ∥d^{(i)}f(x)∥.

Then, 𝔐(U, IR) = C_{b}^{k}(U, IR) is a Banach space, and the axiom (Ax1) is trivially satisfied. Let’s prove that (Ax2) holds with l = k + 1.

Let T : V ⊂ IR^{m} ⟼ IR^{n} be a function of class C^{k+1} defined in the open set V. We need to prove that the map:

(T ∘) : C_{b}^{k}(U, V) ⟼ C_{b}^{k}(U, IR^{n})

is well defined and continuous.

It is easily proven by induction on i that, for i = 0, 1, 2, …, k, there exists a function A:

A : V × ⨁_{j=1}^{i} L_{j}(IR^{d}, IR^{m}) ⊂ ⨁_{j=0}^{i} L_{j}(IR^{d}, IR^{m}) ⟼ L_{i}(IR^{d}, IR^{n})

of class C^{k+1−i} such that:

d^{(i)}(T ∘ f)(x) = A(f(x), df(x), d^{(2)}f(x), …, d^{(i)}f(x)).

Now, let f and g be two functions in C_{b}^{k}(U, IR^{m}) and K a compact subset of

V × ⨁_{j=1}^{i} L_{j}(IR^{d}, IR^{m}) (i ≥ 1) containing the image of (f(x), df(x), …, d^{(i)}f(x))

and of (g(x), dg(x), …, d^{(i)}g(x)). Then, it is:

∥d^{(i)}(T ∘ f) − d^{(i)}(T ∘ g)∥ = ∥A(f, df, …, d^{(i)}f) − A(g, dg, …, d^{(i)}g)∥ ≤

≤ sup_{z∈K} ∥dA(z)∥ · [∥f − g∥ + ∑_{j=1}^{i} ∥d^{(j)}f − d^{(j)}g∥],

which implies easily that (T ∘) is well defined and locally Lipschitz.

We now proceed to the general case C_{b}^{k}(M, IR), where M is a manifold of class C^{k}. We will assume that M is embedded in IR^{d} and that U ⊂ IR^{d} is an open subset containing M. Observe that, in principle, the space C_{b}^{k}(M, IR) does depend on the choice of the embedding of M.

Consider the linear map given by the restriction f ⟼ f|_{M} defined in the Banach space C_{b}^{k}(U, IR) and taking values in the linear space of all functions f : M ⟼ IR. Define C_{b}^{k}(M, IR) as the image of such a map; in this way, C_{b}^{k}(M, IR) becomes isomorfic (as a linear space) to the quotient space:

C_{b}^{k}(U, IR)/C_{b,0}^{k}(U, IR),

where C_{b,0}^{k}(U, IR) denotes the closed subspace of functions vanishing at M. Consider C_{b}^{k}(M, IR) with the induced quotient Banach structure. Now, the axiom (Ax1) holds trivially and (Ax2) follows easily from general properties of the quotient topology.

6.3. A mixed Sobolev space

Let p, q ∈ [1, +∞[ and d ∈ IN be such that:

\frac{1}{p} − \frac{1}{q} = \frac{1}{d}, q > d;

let Ω ⊂ IR^{d} be an open subset. We consider the Banach space H_{p,q}(Ω, IR) given by the completion of the linear space C_{c}^{∞}(IR^{3}, IR) with respect to the norm ∥ · ∥_{p,q} defined by:

∥ϕ∥_{p,q} = ∥dϕ∥_{L^{p}} + ∥dϕ∥_{L^{q}},

where

∥dϕ∥_{L^{r}} = (∫_{Ω} ∥dϕ(x)∥^{r} dx)^{\frac{1}{r}} , r ≥ 1.

By definition of the norm ∥ · ∥_{p,q}, the derivative dϕ belongs to the intersection:

L^{p}(Ω, IR^{d}) ∩ L^{q}(Ω, IR^{d}),

where we identify the target space IR^{d} as the set of linear maps from IR to IR^{d}.

By the inequality of Sobolev, Gagliardo and Nirenberg (see [2]), there exists a positive constant C_{1}, depending on Ω such that, for ϕ ∈ H_{p,q}(Ω, IR), it is:

∥ϕ∥_{L^{q}} ≤ C_{1} · ∥dϕ∥_{L^{p}}.

This implies that we have a continuous immersion H_{p,q}(Ω, IR) ⊂ W^{1,q}(Ω, IR). Then, since q > d, by Morrey’s inequality we also have a continuous immersion:

H_{p,q}(Ω, IR) ⊂ L^{∞}(Ω, IR);  (6.3.1)

moreover there exists a positive constant C_{2} such that, for all ϕ ∈ H_{p,q}(Ω, IR), it is:

|ϕ(x) − ϕ(y)| ≤ C_{2} · ∥dϕ∥_{L^{q}} · |x − y|^{1− \frac{d}{q}} .

Hence, we have a continuous injection of H(Ω, IR) in the space C^{0,α}(Ω, IR) of all Hölder functions with Hölder’s constant α = 1 − \frac{d}{q}.

Finally, by the continuous immersion into L^{∞}, we have that for all ϕ ∈ H_{p,q}(Ω, IR) it is:

lim_{|x|→∞} ϕ(x) = 0,

because every function in H_{p,q}(Ω, IR) is uniform limit of functions with compact support; hence

H_{p,q}(Ω, IR) ⊂ C_{0}(Ω, IR),

with continuous immersion.

So, the Banach space H_{p,q}(Ω, IR) satisfies the axiom (Ax1)′.

Let now U be an open subset of IR^{n} containing 0 and T : U ⟼ IR^{m} be a map of class C^{2} satisfying T (0) = 0.

Then, clearly (T ∘)(C_{c}^{∞}(Ω, U)) ⊆ C_{c}^{2}(Ω, IR^{m}). To prove that (T ∘) admits a continuous extension from H_{p,q}(Ω, U) to H_{p,q}(Ω, IR^{m}) we show that (T ∘) is locally uniformly continuous in C_{c}(Ω, U) with respect to the norm ∥ · ∥_{p,q}.

To see this, let ϕ_{1} ∈ C_{c}^{∞}(Ω, U) be fixed and ϕ_{2} ∈ C_{c}^{∞}(Ω, U) be sufficiently close to ϕ_{2} in the uniform topology so that, for all x ∈ IR^{3}, the segment S_{x} between ϕ_{1}(x) and ϕ_{2}(x) be entirely contained in U. Moreover, let K be any fixed compact subset of U containing the images of ϕ_{1} and ϕ_{2}.

Using the triangle inequality and the Mean Value Theorem one sees easily that the following inequality holds:

(6.3.2) \begin{matrix}∥dT (ϕ_{1}(x)) & [dϕ_{1}(x)] − dT (ϕ_{2}(x))[dϕ_{2}(x)]∥ ≤ \\ ≤ c_{1} · ∥dϕ_{1}(x) − dϕ_{2}(x)∥ + c_{2} · ∥dϕ_{2}(x)∥ · ∥ϕ_{1}(x) − ϕ_{2}(x)∥, \end{matrix}

where

c_{1} = max_{K} ∥dT (x)∥ and c_{2} = max_{S_{x}} ∥d^{2}T (y)∥.

From (6.3.1) and (6.3.2) it follows immediately that (T ∘) is locally uniformly continuous, hence it admits a continuous extension to H_{p,q}(Ω, U), and so H_{p,q}(Ω, IR) satisfies the axiom (Ax2)′ with l = 2.

From Theorem 5.3 it follows that, if N is a C^{k+2}-submanifold of IR^{n}, with k ≥ 2 and 0 ∈ N, then H_{p,q}(Ω, N) has the structure of a Banach submanifold of class C^{k−1} of the Banach space H(Ω, IR^{n}).

The previous example was studied in ([1]) in the case d = 3, p = 2, q = 6 and Ω = IR^{3} and it was the initial motivation for the work presented in this paper.

REFERENCES

[1] V. Benci, F. Giannoni, P. Piccione, A Variational Problem for Manifold Valued Functions, preprint 1997.

[2] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson, Paris– New York, 1983.

[3] J. Eells, A Setting for Global Analysis, Bull. A.M.S. 72, vol. 5 (1966), 751–807.

[4] J. Eells, L. Lemaire, Selected Topics in Harmonic Maps, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics n. 50 (1980)

[5] S. Lang, Differentiable Manifolds, Springer–Verlag, Berlin, 1985.

[6] R. Palais, Foundations of Global Nonlinear Analysis, W. A. Benjamin, 1968.

Address: Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão, 1010 Cidade Universitária, CEP 05508-900 São Paulo, SP Brazil.

e-mail: piccione@ime.usp.br, tausk@ime.usp.br