BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS

DANIEL VICTOR TAUSK

ABSTRACT. We present a method for introducing a Banach manifold structure on sets of maps f :Ω → M, where Ω is a set and M is a manifold. This new approach generalizes and simplifies the classical work developed on [4] and also the more recent work [5].

1. INTRODUCTION The introduction of infinite-dimensional manifold structures on sets of maps is the foun- dation of modern calculus of variations and global analysis. It also plays the central role in concrete applications of infinite-dimensional Morse theory and general critical point theory. In this paper we give conditions under which sets of maps can be endowed with Banach manifold structures. More specifically, we consider the following setup. Let Ω be an arbitrary set and assume that we are given a rule M that assigns to each smooth manifold M a set M(Ω,M) of maps f :Ω → M and a topology on the set M(Ω,M). We assume the validity of eight natural axioms for the rule M and we show that for every manifold M the topological space M(Ω,M) can be endowed with the structure of a Banach manifold. Such structure will be explicitly described in terms of local charts. Moreover, the Banach manifold structure of M(Ω,M) is unique under some naturality conditions. The eight axioms and the detailed construction of the Banach manifold structure on M(Ω,M) is presented in Section 2. In Section 3 some concrete examples where the theory applies are discussed; first, we list a few rules M for which the topological spaces M(Ω,M) can be easily described for arbitrary manifolds M. Then we give a general theorem showing that, given a Banach space E of maps f :Ω → IR satisfying two simple conditions, then there exists a unique rule M satisfying the eight axioms of Section 2 and with M(Ω, IR) = E. Using this theorem we are able to construct several other examples of rules M for which the theory of Section 2 applies. We emphasize that the maps f :Ω → M belonging to our Banach manifolds M(Ω,M) need not be continuous and in fact, in some examples, one does not even need to fix a topology on Ω. For example, for any set Ω, we obtain a Banach manifold structure on the set B(Ω,M) of all maps f :Ω → M with relatively compact image. If Ω is an arbitrary measure space, we also define a Banach manifold based on the space of bounded maps f :Ω → IR with f ∈ Lp(Ω, IR). In what follows we will make a comparison between the constructions presented in this paper and others appearing in the literature. We also sketch the idea behind our construc- tion of the manifold structure of M(Ω,M). It should be pointed out that in this paper we only consider Banach and Hilbert manifold structures, which are in practice more applicable from the point of view of critical point

Date: April 2002. 2000 Mathematics Subject Classification. 46T10, 58D15, 46T05. The author is partially supported by CNPq (Brazil). 1 BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 2 theory. The theory of manifolds modelled on more general locally convex topological vector spaces have been recently developed in detail in [2]. The classical work [4] introduces a Banach manifold structure on a set of sections M(E) of a smooth fiber bundle E over a smooth compact manifold with boundary Ω. The Banach manifold M(E) is modelled on Banach spaces of the form M(ξ), where ξ is a vector bundle neighborhood on E, i.e., ξ is an open subset of E endowed with the structure of a vector bundle over Ω. The rule M that assigns to each vector bundle ξ a Banach space of sections M(ξ) of ξ is supposed to satisfy some axioms. For example, M(ξ) should have a continuous inclusion on the space of continuous sections of ξ (endowed with the compact- open topology) and the left composition maps M(ξ) → M(η) induced by smooth fiber bundle morphisms φ : ξ → η should be continuous. Banach manifold structures for sets of maps f :Ω → M, where M is an arbitrary manifold, are obtained by identifying such maps with sections of the trivial fiber bundle Ω × M → Ω. Following the approach of [4], one would not expect to obtain Banach manifold struc- tures on sets of sections M(E) of a fiber bundle E whose base manifold Ω is not compact. For the case of noncompact bases, the standard literature on the subject tends to present Frechet´ manifold structures; typically, the topology of M(E) is induced by the restriction maps M(E) → M(E|K ), where K ⊂ Ω is a compact domain and M(E|K ) is endowed with a Banach manifold structure. In [5], Banach manifold structures for sets of maps with noncompact domains are stud- ied. The basic example of the theory of [5] is the nonlinear version of the Banach space 0 Cb(Ω, IR) of bounded continuous maps f :Ω → IR defined on a (not necessarily com- pact) topological space Ω, endowed with the sup norm. More precisely, it is introduced 0 a Banach manifold structure on the set Cb(Ω,M) of continuous maps f :Ω → M with relatively compact image on M. The technique for introducing the Banach manifold struc- 0 ture on Cb(Ω,M) is based on an infinite-dimensional version of the rank theorem, which 0 is used to show that Cb(Ω,M) is a smooth (embedded) submanifold of the Banach space 0 n n Cb(Ω, IR ), where M is embedded in IR using Whitney’s theorem. Later, it is shown 0 that the manifold structure of Cb(Ω,M) does not depend on the particular embedding of M in the Euclidean space. We point out that there is no explicit description of a smooth 0 atlas for the Banach manifold Cb(Ω,M) in [5]. From [5] it becomes clear that compactness of the domain Ω is not a crucial property for the introduction of a Banach manifold structure on a set of maps M(Ω,M); indeed, one only needs compactness on the image (or in the closure of the image) of the maps in M(Ω,M). The problem is that this new view is in principle conflicting with the spirit of [4]; namely, if s :Ω → E is a continuous section of a fiber bundle E → Ω then the image of s is always closed in E and homeomorphic to Ω, so that compactness in the image is actually equivalent to compactness in the domain. One is therefore not expected to obtain examples of Banach manifolds of maps with noncompact domain if spaces of maps f :Ω → M are considered as particular cases of spaces of sections of fiber bundles. Any axiomatization for a rule M that leads to the introduction of Banach manifold structures on sets of maps M(Ω,M) (or M(E)) should include some sort of axiom re- quiring continuity of left composition maps f 7→ φ ◦ f (at the very least, one needs continuity of f 7→ φ ◦ f when φ is a smooth diffeomorphism). If, as in [4], one mod- els the Banach manifolds on Banach spaces M(ξ) of sections of a vector bundle ξ → Ω then the natural left composition maps f 7→ φ ◦ f to be considered are the ones induced by smooth fiber bundle morphisms φ : ξ → η (see [4, Axiom §5]). In the case where ξ = Ω × IRm, η = Ω × IRn are trivial vector bundles, such morphisms φ take the form BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 3

Ω × IRm 3 (a, x) 7→ a, φ(a, x)
∈ Ω × IRn and the corresponding left composition map carries f :Ω → IRm to Ω 3 a 7→ φa, f(a)
∈ IRn. If Ω is not compact then, in most examples, such left composition maps are not continuous or even well-defined, i.e., they do not take values in M(Ω, IRn). Instead, one usually has only the continuity of the left composition map f 7→ φ ◦ f where φ is applied only to f(a) and not to a. This gives an- other hint of the fact that it is more natural to work directly with sets of maps f :Ω → M and not with sections of trivial fiber bundles. It should be mentioned that the use of Banach differentiable structures on sets of maps with noncompact domains is indeed relevant. For instance, in the study of Morse homology in [6], one is lead to consider Sobolev spaces of maps of the form H1(IR, M) (see the introduction of [5] for more examples where noncompact domains are used). We emphasize that, differently from [5], the constructions and the main1 results of this paper never use embeddings of the manifolds into Euclidean space. This is possibly a matter of personal taste, but the author feels that this is the most elegant way of dealing with manifolds. Also, the explicit description of coordinate charts for M(Ω,M) is often useful when dealing with spaces of maps in practical applications. Let us now give a sketch of the main ideas behind the construction of the differentiable structure of M(Ω,M). The most tempting way of defining a coordinate chart on M(Ω,M) is to consider the left composition map f 7→ ϕ ◦ f, where ϕ : U ⊂ M → IRn is a coordinate chart on M. This kind of charts are indeed smoothly compatible with each other in the examples we consider, but obviously one cannot expect that they form an atlas for M(Ω,M); namely, there may be maps f ∈ M(Ω,M) whose image is not contained in the domain of a chart of M. Since we only consider maps f with relatively compact image, one can cover the image of f with a finite number of coordinate charts ϕi, i = 1, . . . , r, but there is no visible way of combining the charts ϕi, i = 1, . . . , r into a chart for M(Ω,M) around f. Instead, let us take a look at the vector bundle neighborhoods of [4]. As explained before in this introduction, a vector bundle neighborhood ξ on a fiber bundle E → Ω is an open set ξ ⊂ E which has the structure of a vector bundle over Ω. Obviously, in our case, Ω may not even be a topological space, so we shouldn’t talk about fiber bundles over Ω; but let us just for the moment assume that Ω is a manifold. For each a ∈ Ω, the fiber ξa of ξ over a is an open subset of the fiber Ea of E over a; moreover, ξa is endowed with the structure of a real finite-dimensional vector space. Just for psychological reasons, it seems simpler to picture this situation in terms of a smooth diffeomorphism ϕa : Va → ξa, where Va ⊂ Ea is open and ξa is a real finite-dimensional vector space. Now ϕa is just a chart on Ea; in the case that E = Ω × M is a trivial bundle, the vector bundle neighborhood ξ can be thought of as a family of charts (ϕa)a∈Ω on M, where the domain of ϕa is an open subset Va of M (depending on a) and the counter- domain of ϕa is a vector space ξa (also depending on a). So, rather than trying to cover the image of a map f :Ω → M with a finite number of charts, we cover it with a continuous family of charts (ϕa)a∈Ω, where f(a) belongs to the domain of ϕa for every a ∈ Ω. This yields a chart on M(Ω,M) around f taking values in the space of sections of a vector bundle. Now, the key observation here is that the index set for the family of charts (ϕa)a∈Ω doesn’t need to have anything to do with the domain Ω of the maps f. Instead, we consider an arbitrary smooth manifold X to parameterize the charts ϕa and we consider an arbitrary vector bundle ξ over X playing the role of the old vector bundle neighborhood. We then consider a smooth diffeomorphism ϕ : V → Ve between open subsets V ⊂ X × M and

1We did use the existence of embeddings into Euclidean space occasionally in some less important remarks. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 4

Ve ⊂ ξ such that for every x ∈ X, the map ϕx = ϕ(x, ·) carries Vx = V ∩ {x} × M to Vex = Ve ∩ ξx. Thus (ϕx)x∈X is now a parameterized family of charts on M. In order to complete the construction of the chart on M(Ω,M) around f :Ω → M, we need a map σ :Ω → X that tells us for each a ∈ Ω, which of the charts ϕx should be used around f(a); more explicitly, the chart on M(Ω,M) is of the form f 7→ ϕ ◦ (σ, f) and it takes values on the space of sections s :Ω → ξ of ξ along σ. It may sound a bit surprising that the construction of a chart around f does not use any sort of continuity of f and does not use approximations by smooth maps as in [4].

2. THE GENERAL AXIOMATIZATION Given sets A, B we will denote by BA the set of all maps from A to B. Throughout this section we will consider fixed a set Ω and a rule M that assigns to each manifold M a subset M(Ω,M) of M Ω and a topology on the set M(Ω,M). By a manifold we will mean a smooth finite-dimensional real manifold whose topology is Hausdorff and second countable, where smooth means “of class C∞”. Below we will list a few axioms concerning the rule M that will allow us to construct a Banach manifold structure on the topological space M(Ω,M).

Axiom A0. There exists a manifold M0 for which M(Ω,M0) is nonempty. Axiom A1. Given manifolds M, N and a smooth map φ : M → N then φ◦f ∈ M(Ω,N) for all f ∈ M(Ω,M); moreover, the left composition map: LC(φ): M(Ω,M) −→ M(Ω,N) defined by LC(φ)(f) = φ ◦ f is continuous. Obviously axiom (A1) implies that if φ : M → N is a smooth diffeomorphism then LC(φ) is a homeomorphism. Also, from axioms (A0) and (A1) we obtain that M(Ω,M) contains all constant maps; to see that, simply evaluate LC(φ) in an arbitrary element of M(Ω,M0), where φ : M0 → M is an arbitrary constant map.

Axiom A2. Let M1, M2 be manifolds and denote by pr1, pr2 the projections of the product M1 × M2. The map:
(2.1) LC(pr1), LC(pr2) : M(Ω,M1 × M2) −→ M(Ω,M1) × M(Ω,M2) is a homeomorphism, where the counter-domain of (2.1) is endowed with the standard product topology. Obviously one can show by induction a version of axiom (A2) for arbitrary finite prod- ucts of manifolds. Axiom A3. For any manifold M the elements of M(Ω,M) have relatively compact image in M, i.e., the set Im(f) is compact for all f ∈ M(Ω,M). Axiom A4. Let M be a manifold and U ⊂ M be an open subset. If f ∈ M(Ω,M) and Im(f) ⊂ U then the map f :Ω → U is in M(Ω,U). If U is open in M then axiom (A1) implies that M(Ω,U) is a subset of M(Ω,M), provided that we identify U Ω with a subset of M Ω in the obvious way. Moreover, axioms (A3) and (A4) imply that: (2.2) M(Ω,U) = f ∈ M(Ω,M): Im(f) ⊂ U . BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 5

For later use we observe that (2.2) implies: (2.3) M(Ω,U ∩ V ) = M(Ω,U) ∩ M(Ω,V ), for any open subsets U, V ⊂ M. Also, axioms (A3) and (A4) imply that: (2.4) MΩ, φ−1(U)
= LC(φ)−1M(Ω,U)
, for every smooth map φ : M → N and every open subset U ⊂ N. Axiom A5. By identifying U Ω with a subset of M Ω then M(Ω,U) is open in M(Ω,M) and has the topology induced from M(Ω,M). Axiom A6. Given a manifold M and a point a ∈ Ω then the evaluation map:

(2.5) evala : M(Ω,M) −→ M given by evala(f) = f(a) is continuous. Observe that axiom (A6) means that the topology of M(Ω,M) is finer than pointwise convergence topology, i.e., the topology induced from the product topology on M Ω. In particular, all spaces M(Ω,M) are Hausdorff. Before stating the last axiom, we prove the following: 2.1. Lemma. If E is a finite-dimensional real vector space (regarded as a manifold in the canonical way) then the set M(Ω,E) is a subspace of the vector space EΩ. For t ∈ IR, denote by ct :Ω → IR the constant map equal to t; assuming that the map:

(2.6) IR 3 t 7−→ ct ∈ M(Ω, IR) is continuous then M(Ω,E) is a topological vector space, i.e., the vector space operations of M(Ω,E) are continuous. Proof. We know that M(Ω,E) contains the constant maps, so it contains the identically zero map. Moreover, by axiom (A2), M(Ω,E × E) can be identified with the product M(Ω,E) × M(Ω,E); since the sum map E × E 3 (v, w) 7→ v + w ∈ E is smooth, axiom (A1) implies that M(Ω,E) is closed under addition and that the sum of M(Ω,E) is continuous. For fixed t ∈ IR, the homotety E 3 v 7→ tv ∈ E is smooth, and thus axiom (A1) implies that M(Ω,E) is closed under scalar multiplication. Finally, by axioms (A1) and (A2), the map M(Ω, IR) × M(Ω,E) → M(Ω,E) induced by scalar multiplication is continuous; thus, if (2.6) is continuous, then the scalar multiplication of M(Ω,E) is also continuous. 

Axiom A7. The real vector space M(Ω, IR) is Banachble, i.e., there exists a norm on M(Ω, IR) that induces its topology and that makes it into a Banach space. Obviously axiom (A2) implies that M(Ω, IRn) is linearly homeomorphic to the topo- L n logical direct sum n M(Ω, IR) and thus M(Ω, IR ) is also a Banachble space, for all n. More generally, M(Ω,E) is a Banachble space for every real finite-dimensional vector space E, by axiom (A1). 2.2. Lemma. Given manifolds M, N and a smooth embedding φ : N → M then the map LC(φ): M(Ω,N) → M(Ω,M) is a homeomorphism onto its image, which is given by: (2.7) ImLC(φ)
= f ∈ M(Ω,M): Im(f) ⊂ Im(φ) . Moreover, if Im(φ) is closed in M then ImLC(φ)
is closed in M(Ω,M). BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 6

Proof. Since φ : N → Im(φ) is a smooth diffeomorphism, by axiom (A1), we can assume without loss of generality that N is a submanifold of M and that φ is the inclusion map. We then identify M(Ω,N) with a subset of M(Ω,M), being LC(φ) the inclusion map. Using a tubular neighborhood of N in M we can find an open set U ⊂ M containing N and a smooth retraction r : U → N, i.e., r|N = IdN . Then LC(r) is a continuous left inverse of the inclusion of M(Ω,N) in M(Ω,U); observing that, by axiom (A5), M(Ω,U) has the topology induced from M(Ω,M), we conclude that M(Ω,N) has the topology induced from M(Ω,M), i.e., LC(φ) is a homeomorphism onto its image. The inclusion of the lefthand side of (2.7) into the righthand side of (2.7) follows from axiom (A3). Moreover, if f ∈ M(Ω,M) and Im(f) ⊂ N then f ∈ M(Ω,U), by axiom (A4) and thus f = LC(r)(f) is in M(Ω,N). This proves (2.7). Finally, the fact that M(Ω,N) is closed in M(Ω,M) if N is closed in M follows from axiom (A6).  Let π : ξ → X be a smooth vector bundle over a manifold X. Given σ ∈ M(Ω,X) we set: M(Ω, ξ; σ) = s ∈ M(Ω, ξ): π ◦ s = σ , and we endow M(Ω, ξ; σ) with the topology induced by M(Ω, ξ). If U ⊂ ξ is open then we also write: M(Ω,U; σ) = M(Ω,U) ∩ M(Ω, ξ; σ), and we endow M(Ω,U; σ) with the topology induced by M(Ω, ξ), which, by axiom (A5), coincides with the topology induced by M(Ω,U). Moreover, M(Ω,U; σ) is open in M(Ω, ξ; σ). 2.3. Lemma. If π : ξ → X is a smooth vector bundle over a manifold X and σ is in M(Ω,X) then M(Ω, ξ; σ) is a subspace of the vector space of all maps s :Ω → ξ with π ◦ s = σ. Moreover, M(Ω, ξ; σ) is a Banachble space.

Proof. By axiom (A3) we can cover Im(σ) with a finite number of open sets Ui ⊂ X, i = 1, . . . , r, such that ξ is trivial over Ui for each i. Denote by n the dimension of the −1 n fibers of ξ and for each i = 1, . . . , r let φi : π (Ui) → IR be a smooth map whose r restriction to each fiber is an isomorphism. Let (λi)i=1 be a smooth partition of unity on Sr the open set U = i=1 Ui such that supp(λi) ⊂ Ui for all i. Consider the smooth map M φ : π−1(U) −→ IRn =∼ IRrn r −1 −1 whose i-th coordinate equals (λi ◦ π)φi on π (Ui) and equals zero on π (U \ Ui), for i = 1, . . . , r. Then (π, φ): π−1(U) → U × IRrn is a smooth vector bundle isomorphism −1 rn from π (U) = ξ|U onto a vector subbundle ξe of the trivial bundle U × IR . Observe that if s ∈ M(Ω, ξ; σ) then Im(s) ⊂ ξ|U and thus, by axioms (A4) and (A5), we have rn M(Ω, ξ; σ) = M(Ω, ξ|U ; σ). Since ξe is a closed submanifold of U × IR , Lemma 2.2 implies that LC(π, φ) is a linear homeomorphism between M(Ω, ξ|U ; σ) and the closed rn ∼ rn subspace M(Ω, ξe; σ) of the Banachble space M(Ω,U × IR ; σ) = M(Ω, IR ).  In order to construct a smooth atlas on the spaces M(Ω,M), we will have to prove the smoothness of certain left composition maps on Banachble spaces of the form M(Ω, ξ; σ). To that aim, we will employ a general lemma that allows one to establish differentiability of maps between Banach spaces. 2.4. Definition. Let E be a Banachble space. A separating set of continuous linear maps for E is a set Λ of continuous linear maps λ : E → F , where F is a Banachble space that may depend on λ, such that for every nonzero v ∈ E there exists λ ∈ Λ with λ(v) 6= 0. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 7

Given Banachble spaces E, F , we denote by Lin(E,F ) the Banachble space of contin- uous linear maps from E to F . 2.5. Lemma (weak differentiation principle). Let E, F be Banachble spaces, U ⊂ E an open subset, f : U → F a map and Λ a separating set of continuous linear maps for F . If there exists a continuous map g : U → Lin(E,F ) such that: d (2.8) (λ ◦ f)(x + tv) = λg(x)v
, dt t=0 for all x ∈ U, v ∈ E, λ ∈ Λ then f is of class C1 and df = g.

Proof. See [5, Proposition 3.2].  Let X be a manifold and π1 : ξ1 → X, π2 : ξ2 → X be smooth vector bundles over X. We denote by ξ1 ⊕ ξ2 the Whitney sum of ξ1 and ξ2, i.e., the vector bundle 1 2 over X whose fiber over x ∈ X is ξx ⊕ ξx. There exists an obvious diffeomorphism between ξ1 ⊕ ξ2 and the closed submanifold of the product ξ1 × ξ2 consisting of pairs (v, w) ∈ ξ1 × ξ2 with π1(v) = π2(w). Thus, using Lemma 2.2 and axiom (A2) we obtain a linear homeomorphism: (2.9) M(Ω, ξ1 ⊕ ξ2; σ) =∼ M(Ω, ξ1; σ) ⊕ M(Ω, ξ2; σ), for any σ ∈ M(Ω,X). Denote by Lin(ξ1, ξ2) the vector bundle over X whose fiber over x ∈ X is the space 1 2 1 2 Lin(ξx, ξx) of linear maps from ξx to ξx. Consider the smooth map: C : Lin(ξ1, ξ2) ⊕ ξ1 −→ ξ2 1 2 1 defined by C(T, v) = T (v), for all T ∈ Lin(ξx, ξx), v ∈ ξx, x ∈ X. By axiom (A1), the map LC(C) is continuous and using the identification given in (2.9) we obtain a continuous bilinear map: LC(C): MΩ, Lin(ξ1, ξ2); σ
× M(Ω, ξ1; σ) −→ M(Ω, ξ2; σ). The continuous bilinear map LC(C) above then induces in a natural way a continuous linear map: (2.10) O : MΩ, Lin(ξ1, ξ2); σ
−→ LinM(Ω, ξ1; σ), M(Ω, ξ2; σ)
; more explicitly, we have O(T )(s)(a) = T (a)s(a), for every T ∈ MΩ, Lin(ξ1, ξ2); σ
, s ∈ M(Ω, ξ1; σ) and a ∈ Ω. The construction above will be used in the proof of Lemma 2.6 below. Recall that, given smooth vector bundles π1 : ξ1 → X, π2 : ξ2 → X over a manifold X then a map φ defined on a subset of ξ1, taking values in ξ2 is called fiber preserving if π2φ(v)
= π1(v), for all v in the domain of φ. 2.6. Lemma. Let π1 : ξ1 → X, π2 : ξ2 → X be smooth vector bundles over a manifold X and let φ : U → ξ2 be a smooth fiber preserving map defined on an open subset U ⊂ ξ1. Given σ ∈ M(Ω,X) then: LC(φ): M(Ω,U; σ) −→ M(Ω, ξ2; σ) is a smooth map on the open subset M(Ω,U; σ) of the Banachble space M(Ω, ξ1; σ).

Proof. Denote by Fφ the fiber derivative of φ which is the smooth fiber preserving map: 1 2 Fφ : U −→ Lin(ξ , ξ ) BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 8

1 defined by φ(v) = d(φ| 1 )(v), for all v ∈ ξ ∩ U, x ∈ X. Our strategy is to use F ξx∩U x Lemma 2.5 to show that LC(φ) is of class C1 and that:
d LC(φ) = O ◦ LC(Fφ). The smoothness of LC(φ) will then follow by induction. We already know that O is continuous and that LC(Fφ) is continuous, by axiom (A1). For each a ∈ Ω, denote by: 2 2 evala : M(Ω, ξ ; σ) −→ ξσ(a) the map of evaluation at a. We know from axiom (A6) that evala is a continuous linear  map and then obviously Λ = evala : a ∈ Ω is a separating set of continuous linear maps for M(Ω, ξ2; σ). The verification of the hypothesis (2.8) on Lemma 2.5, with f = LC(φ), g = O ◦ LC(Fφ), is now straightforward.  We will now consider a fixed manifold M and we will construct a smooth atlas for M(Ω,M). Let π : ξ → X be a smooth vector bundle over a manifold X and let:

ϕ : X × M ⊃ V −→ Ve ⊂ ξ be a smooth diffeomorphism, where V is open in X × M and Ve is open in ξ; we assume in addition that ϕ is fiber preserving in the sense that π ◦ ϕ = pr1|V , where pr1 denotes the first projection of the product X × M. Given σ ∈ M(Ω,X) we write: (2.11) M(Ω,M; σ, V ) = f ∈ M(Ω,M):(σ, f) ∈ M(Ω,V ) ⊂ M(Ω,M), and we consider the map:

(2.12) LC(ϕ; σ): M(Ω,M; σ, V ) −→ M(Ω, Ve; σ) ⊂ M(Ω, ξ; σ), defined by LC(ϕ; σ)(f) = ϕ ◦ (σ, f). The set M(Ω,M; σ, V ) is open in M(Ω,V ), by axioms (A2) and (A5); moreover, M(Ω, Ve; σ) is open in the Banachble space M(Ω, ξ; σ) and LC(ϕ; σ) is a homeomorphism, by axiom (A1). Thus LC(ϕ; σ) is a (topological) local chart in the topological space M(Ω,M). We will call X the auxiliary manifold and σ the auxiliary map corresponding to the chart LC(ϕ; σ). Our goal now is to show that the local charts (2.12) form a smooth atlas on M(Ω,M). 2.7. Lemma. Let π : ξ → X, π0 : η → Y be smooth vector bundles over manifolds X, Y and let σ ∈ M(Ω,X), τ ∈ M(Ω,Y ) be fixed. Given smooth fiber preserving diffeomorphisms:

ϕ : X × M ⊃ V −→ Ve ⊂ ξ, ψ : Y × M ⊃ W −→ Wf ⊂ η between open sets V , Ve, W , Wf then the local charts LC(ϕ; σ) and LC(ψ; τ) on M(Ω,M) are smoothly compatible, i.e., the transition map LC(ψ; τ) ◦ LC(ϕ; σ)−1 is a smooth dif- feomorphism between open sets. Proof. The strategy of the proof is two modify the charts LC(ϕ; σ) and LC(ψ; τ) so that they both correspond to the same auxiliary manifold X × Y and the same auxiliary map (σ, τ) ∈ M(Ω,X × Y ). To this aim, consider the smooth vector bundles: 0 π × IdY : ξ × Y −→ X × Y, IdX × π : X × η −→ X × Y over the manifold X × Y . By axiom (A2), we have obvious linear homeomorphisms: M(Ω, ξ; σ) =∼ MΩ, ξ × Y ;(σ, τ)
, M(Ω, η; τ) =∼ MΩ,X × η;(σ, τ)
, BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 9 where (σ, τ) ∈ M(Ω,X ×Y ). By taking restrictions of the linear homeomorphisms above we obtain the following homeomorphisms:

(2.13) M(Ω, Ve; σ) =∼ M Ω, Ve × Y ;(σ, τ) , M(Ω, Wf; τ) =∼ M Ω,X × Wf;(σ, τ) . Also, by axiom (A2), we have (recall (2.11)): (2.14) M(Ω,M; σ, V ) = MΩ,M;(σ, τ), s(Y × V )
, (2.15) M(Ω,M; τ, W ) = MΩ,M;(σ, τ),X × W
, where s : Y × X × M → X × Y × M is the map that swaps the first two coordinates. Consider the smooth fiber preserving diffeomorphisms:

ϕ : X × Y × M ⊃ s(Y × V ) −→ Ve × Y ⊂ ξ × Y, ψ : X × Y × M ⊃ X × W −→ X × Wf ⊂ X × η, defined by ϕ(x, y, m) = ϕ(x, m), y
, ψ(x, y, m) = x, ψ(y, m)
. Using the identities (2.14), (2.15) and the identifications (2.13), the charts LC(ϕ; σ) and LC(ψ; τ) are identi- fied respectively with the charts:


(2.16) LC ϕ;(σ, τ) : M Ω,M;(σ, τ), s(Y × V ) −→ M Ω, Ve × Y ;(σ, τ) ,


(2.17) LC ψ;(σ, τ) : M Ω,M;(σ, τ),X × W ) −→ M Ω,X × Wf;(σ, τ) . By (2.3), the intersection of the domains of the charts above is MΩ,M;(σ, τ),Z
, where: Z = s(Y × V ) ∩ (X × W ) ⊂ X × Y × M. Hence, the transition map between the charts LCϕ;(σ, τ)
and LCψ;(σ, τ)
, which, up to the identifications (2.13), is equal to the transition map between LC(ϕ; σ) and LC(ψ; τ), is given by: −1 LCψ ◦ ϕ ): MΩ, ϕ(Z); (σ, τ)
−→ MΩ, ψ(Z); (σ, τ)
;

−1 since ψ ◦ ϕ : ϕ(Z) → ψ(Z) is a smooth fiber preserving diffeomorphism, Lemma 2.6 implies that the map above is a smooth diffeomorphism between open sets, which proves that LC(ϕ; σ) and LC(ψ; τ) are smoothly compatible.  A topological space X is called hereditarily paracompact if every subspace of X is paracompact (or, equivalently, if every open subspace of X is paracompact). Under our conventions, all manifolds are hereditarily paracompact. We have the following: 2.8. Lemma. Let f : X → Y be a local homeomorphism, where X , Y are topological spaces, with Y Hausdorff and hereditarily paracompact. If S ⊂ X is a subset2 such that f|S : S → f(S) is a homeomorphism then there exists an open set Z ⊂ X containing S such that f|Z : Z → f(Z) is a homeomorphism. Proof. The proof follows a standard argument that is used in some proofs on the existence of tubular neighborhoods (see for instance [3, §5, Chapter IV]).  2.9. Proposition. The local charts of the form (2.12) form a smooth atlas on the topologi- cal space M(Ω,M).

2Actually, if one assumes that f(S) be closed in Y then it would be sufficient to assume that Y be Hausdorff and paracompact, rather than hereditarily paracompact. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 10

Proof. By Lemma 2.7, it suffices to find for every f ∈ M(Ω,M) a chart of the form (2.12) on M(Ω,M) whose domain contains f. Choose an arbitrary Riemannian metric on M (or an arbitrary connection on TM) and denote by exp the corresponding exponential map, which is a smooth M-valued map on an open subset D of TM. If π : TM → M denotes the projection then, by the inverse function theorem, the map (π, exp) : D → M × M is a smooth local diffeomorphism on an open subset of D containing the zero section of TM; thus, by Lemma 2.8, there exists an open set Ve ⊂ TM containing the zero section which is mapped diffeomorphically by (π, exp) onto an open subset V ⊂ M × M containing the diagonal. The desired chart LC(ϕ; σ) is now obtained by taking ξ = TM, X = M, −1 ϕ = (π, exp) : V → Ve and σ = f.  2.10. Remark. As we have already observed, axiom (A6) implies that the space M(Ω,M) is Hausdorff. Actually, since M can be embedded in IRn for some n by Whitney’s theorem, Lemma 2.2 implies that M(Ω,M) is homeomorphic to a subspace of the Banachble space M(Ω, IRn). Thus M(Ω,M) is T4, metrizable and hereditarily paracompact. Moreover, M(Ω,M) is second countable if the Banachble space M(Ω, IR) is separable. 2.11. Remark. Obviously if the Banachble space M(Ω, IR) is Hilbertable then the proof of Lemma 2.3 shows that the spaces M(Ω, ξ; σ) are also Hilbertable and therefore M(Ω,M) is a Hilbert manifold for any manifold M. From now on, on this section, we will assume that the spaces M(Ω,M) are endowed with the Banach manifold structure defined by the charts (2.12) and we will prove a few basic results about such manifold structure. The next two propositions are rather trivial though important for the completeness of the theory. 2.12. Proposition. If E is a finite-dimensional real vector space (regarded as a mani- fold in the canonical way) then the Banachble space M(Ω,E) has its canonical manifold structure, i.e., the manifold structure induced by the atlas containing the identity map of M(Ω,E). Proof. Let X be a one point (zero-dimensional) manifold, ξ = E → X be the trivial bundle over X whose unique fiber is E and let σ ∈ M(Ω,X) be the unique constant map. The chart LC(ϕ; σ): M(Ω,E; σ, X × E) = M(Ω,E) → M(Ω, ξ; σ) = M(Ω,E) induced by the obvious diffeomorphism ϕ : X × E → ξ is equal to the identity map.  2.13. Proposition. If M is a manifold and U ⊂ M is an open subset then M(Ω,U) is an open submanifold of M(Ω,M). Proof. By axiom (A5), M(Ω,U) is open in M(Ω,M). Moreover, the charts we have defined for M(Ω,U) are also charts for M(Ω,M).  2.14. Proposition. Given manifolds M, N and a smooth map φ : M → N then the left composition map LC(φ): M(Ω,M) → M(Ω,N) is smooth. Proof. Let f ∈ M(Ω,M) be fixed and choose local charts: LC(ϕ; σ): M(Ω,M; σ, V ) −→ M(Ω, Ve; σ), LC(ψ; τ): M(Ω,N; τ, W ) −→ M(Ω, Wf; τ), whose domains contain respectively f and φ ◦ f. As usual, the definition of the charts above involve manifolds X, Y , maps σ ∈ M(Ω,X), τ ∈ M(Ω,Y ), smooth vector bundles ξ → X, η → Y , open sets V ⊂ X × M, W ⊂ Y × N, Ve ⊂ ξ, Wf ⊂ η and smooth BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 11

fiber preserving diffeomorphisms ϕ : V → Ve, ψ : W → Wf. Now we proceed as in the proof of Lemma 2.7 in order to modify the charts LC(ϕ; σ) and LC(ψ; τ) so that they correspond to the same auxiliary manifold X × Y and the same auxiliary map (σ, τ). We will then obtain charts LCϕ;(σ, τ)
and LCψ;(σ, τ)
similar to (2.16) and (2.17); here the domain of ϕ is open in X × Y × M while the domain of ψ is open in X × Y × N. The coordinate representation of the map LC(φ) with respect to the charts LCϕ;(σ, τ)
and LCψ;(σ, τ)
is given by (see also (2.4)): −1 −1 LCψ;(σ, τ)
◦ LC(φ) ◦ LCϕ;(σ, τ)
= LCψ ◦ φ ◦ ϕ
,

−1 where φ = Id × Id × φ : X × Y × M → X × Y × N. Since ψ ◦ φ ◦ ϕ is a smooth fiber preserving map between open subsets of the vector bundles ξ × Y and X × η, the conclusion follows from Lemma 2.6.  2.15. Corollary. Given manifolds M, N and a smooth diffeomorphism φ : M → N then also LC(φ): M(Ω,M) → M(Ω,N) is a smooth diffeomorphism.  2.16. Corollary. Given manifolds M, N and a smooth embedding φ : N → M then the left composition map LC(φ): M(Ω,N) → M(Ω,M) is a smooth embedding. Proof. As in the proof of Lemma 2.2, we can find an open subset U ⊂ M containing Im(φ) and a smooth left inverse r : U → N for φ : N → U. Then, by Proposition 2.14, LC(r) is a smooth left inverse of LC(φ): M(Ω,N) → M(Ω,U). This implies that LC(φ) is a smooth embedding in M(Ω,U). The conclusion now follows from Proposition 2.13.  We finish the section by showing the usual identification between the tangent bundle T M(Ω,M) and M(Ω,TM). 2.17. Proposition. For every manifold M and every a ∈ Ω, the evaluation map (2.5) is smooth. Given f ∈ M(Ω,M) and v ∈ Tf M(Ω,M) then the map vˆ :Ω → TM defined by:

vˆ(a) = d(evala)(f)v ∈ Tf(a)M, a ∈ Ω, is in M(Ω,TM). Moreover, the map: (2.18) T M(Ω,M) 3 v 7−→ vˆ ∈ M(Ω,TM) is a smooth diffeomorphism. Proof. Let f ∈ M(Ω,M) be fixed and consider a chart of the form (2.12) on M(Ω,M) whose domain contains f. Given a ∈ Ω then the fiber preserving smooth diffeomorphism: ϕ : X × M ⊃ V −→ Ve ⊂ ξ
induces a chart ϕ σ(a), · around f(a) on M, taking values in ξσ(a). The coordinate
representation of evala with respect to the charts LC(ϕ; σ) and ϕ σ(a), · is simply the restriction to M(Ω, Ve; σ) of the evaluation map:

M(Ω, ξ; σ) 3 s 7−→ s(a) ∈ ξσ(a), which is obviously linear and continuous by axiom (A6). Thus evala is smooth and for v ∈ Tf M(Ω,M) we have:
  (2.19) ∂2ϕ σ(a), f(a) vˆ(a) = d LC(ϕ; σ)(f)v (a), where ∂2ϕ denotes differentiation of the map ϕ with respect to the variable in M. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 12

Denoting by π : TM → M the canonical projection and by pr1 : ξ ⊕ ξ → ξ the projec- tion onto the first component then ϕ induces a fiber preserving smooth diffeomorphism: −1 −1 Fϕ : X × TM ⊃ (Id × π) (V ) −→ pr1 (Ve) ⊂ ξ ⊕ ξ, defined by:
Fϕ(x, z) = ϕ(x, m), ∂2ϕ(x, m)z ∈ ξx ⊕ ξx, for all (x, z) ∈ X × TM with (x, m) ∈ V , where m = π(z) ∈ M. The map Fϕ induces the chart: −1
−1 LC(Fϕ; σ): M Ω,TM; σ, (Id × π) (V ) −→ M Ω, pr1 (Ve); σ) ⊂ M(Ω, ξ ⊕ ξ; σ) on M(Ω,TM). Moreover, the chart LC(ϕ; σ) on M(Ω,M) induces the chart:
LC(ϕ; σ) ◦ p, d LC(ϕ; σ) : T M(Ω,M; σ, V ) −→ M(Ω, Ve; σ) × M(Ω, ξ; σ) on T M(Ω,M), where p : T M(Ω,M) → M(Ω,M) denotes the canonical projection. Identifying M(Ω, ξ ⊕ ξ; σ) with M(Ω, ξ; σ) ⊕ M(Ω, ξ; σ) (recall (2.9)) then by (2.19) the coordinate representation of (2.18) with respect to the charts LC(ϕ; σ) ◦ p, d LC(ϕ; σ)
and LC(Fϕ; σ) is simply the identity map of M(Ω, Ve; σ) × M(Ω, ξ; σ). This shows at the same time that vˆ ∈ M(Ω,TM) and that (2.18) is a smooth diffeomorphism.  2.18. Remark. It is possible to generalize the theory of this section to include spaces of maps M(Ω,M) where M is infinite-dimensional. For instance, one can allow M to belong to the class of Hausdorff paracompact Banach manifolds modelled on a class of Banach spaces that admit a nonzero real valued smooth map with bounded support (for instance, Hilbert spaces). In this case one has to strengthen axiom (A7), so that M(Ω,E) is Banach- ble for every Banach space E in the class under consideration. 2.19. Remark. Given a rule M satisfying axioms (A0)—(A7) then the manifold struc- ture in the topological spaces M(Ω,M) is unique if one assumes the validity of Proposi- tions 2.12, 2.13 and 2.14. Namely, the validity of such propositions implies the validity of Corollary 2.16; thus, if one chooses a smooth embedding φ : M → IRn into Eu- clidean space then LC(φ) must be a smooth embedding of M(Ω,M) into the Banachble space M(Ω, IRn). But there can be at most one manifold structure on M(Ω,M) for which LC(φ) is a smooth embedding.

3. CONCRETE EXAMPLES In this section we present several concrete examples of rules M that satisfy axioms (A0)—(A7) of Section 2. We start by listing some simple examples where the topological space M(Ω,M) can be easily described for every manifold M. Then, we present a method for obtaining the spaces M(Ω,M) from a prescribed Banachble space E = M(Ω, IR) satisfying two simple properties. 3.1. Example. Let Ω be an arbitrary set and for each manifold M let: M(Ω,M) = B(Ω,M) be the space of all maps f :Ω → M with relatively compact image. Choose a metric d on the manifold M compatible with its topology and consider B(Ω,M) endowed with the uniform convergence topology. If φ :(M, d) → (N, d0) is a continuous map then it is easy to see that the map LC(φ): B(Ω,M) → B(Ω,N) is continuous. This implies in particular that the topology on B(Ω,M) does not depend on the metric d, so that we indeed have a topological space B(Ω,M) associated to each manifold M. The verification of the BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 13 axioms (A0)—(A7) is straightforward. For axiom (A7), observe that B(Ω, IR) is simply the space of bounded maps f :Ω → IR, whose topology can be induced by the usual sup norm kfksup = supa∈Ω f(a) . 3.2. Example. Let (Ω, A) be a measurable space, i.e., Ω is a set and A is a σ-algebra of subsets of Ω. A map defined on Ω and taking values in a given topological space is called measurable if the inverse image of every open set is in A. For each manifold M, let M(Ω,M) = Mb(Ω,M) be the set of all measurable maps f :Ω → M with relatively compact image; consider Mb(Ω,M) endowed with the topology induced by B(Ω,M). One easily checks axioms (A0)—(A7). For axiom (A7), observe that Mb(Ω, IR) is a closed subspace of B(Ω, IR). 0 3.3. Example. Let Ω be an arbitrary topological space and let M(Ω,M) = Cb(Ω,M) be the set of all continuous maps f :Ω → M with relatively compact image; consider 0 Cb(Ω,M) endowed with the topology induced by B(Ω,M). Axioms (A0)—(A7) are 0 easily checked. Observe that Cb(Ω, IR) is the Banachble space of bounded real valued continuous maps on Ω, whose topology is induced by the sup norm. 3.4. Example. Let Ω be an open subset of IRm and k ≥ 1 be fixed. Given a manifold M, the k-th jet bundle J k(IRm,M) is a fiber bundle over M constructed as follows; for each x ∈ M, the fiber of J k(IRm,M) over x is the set of equivalence classes of M-valued maps f of class Ck defined in a neighborhood of the origin in IRm and with f(0) = x. The equivalence relation is f1 ∼ f2 iff f1 and f2 have the same Taylor polynomial of order k at the origin, when some coordinate chart of M is used around x. There are well-known natural local trivializations of J k(IRm,M) induced by local charts of M. Given a map f :Ω → M of class Ck we define the k-th jet of f as the continuous map J k(f):Ω → J k(IRm,M) such that J k(f)(a) ∈ J k(IRm,M) is the equivalence m m k class of f ◦ ta, where ta : IR → IR denotes the translation by a. We define Cb (Ω,M) to be the set of maps f :Ω → M of class Ck such that J k(f) has relatively compact image k m k k m
in J (IR ,M). The topology on Cb (Ω,M) will be induced from B Ω,J (IR ,M) by k k the map f 7→ J (f). It is easy to see that M = Cb satisfies axioms (A0)—(A7). Observe k k that Cb (Ω, IR) is the Banachble space of bounded maps f :Ω → IR of class C having bounded partial derivatives up to order k, whose topology is induced by the standard Ck norm: X kfk = k∂λfksup, |λ|≤k m Pm where λ = (λ1, . . . , λm) ∈ IN denotes a multi-index and |λ| = i=1 λi. This example do not generalize directly to the case that Ω is a manifold. In order to k make sense of the space Cb (Ω,M) in this case, one needs for instance a connection and a Riemannian metric on Ω (this will be dealt with in Example 3.11 below). We now present a more systematic method for producing examples of rules M satisfying axioms (A0)—(A7). The idea is the following. We start with an arbitrary set Ω and a Banachble space E of maps f :Ω → IR satisfying a suitable property and we will then show how to construct a rule M satisfying axioms (A0)—(A7) for which M(Ω, IR) = E. Let Ω be an arbitrary set. As in Example 3.1, we denote by B(Ω, IRn) the Banach space of all bounded maps f :Ω → IRn endowed with the sup norm. We give the following: 3.5. Definition. A Banachble space of bounded maps on Ω is a Banachble space E which is a vector subspace of B(Ω, IR) and such that the inclusion map E ,→ B(Ω, IR) is con- n L tinuous. For all n ≥ 1, we identify the Banachble space E = n E with the subspace of BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 14

n n B(Ω, IR ) consisting of those maps f :Ω → IR all of whose coordinates fi :Ω → IR are in E. We say that E has the left composition property if for every n ≥ 1 and every smooth map φ : IRn → IR the left composition map: LC(φ): En 3 f 7−→ φ ◦ f ∈ E is well-defined3 and continuous. We insist on calling E a Banachble space, rather than a Banach space, to emphasize that only the topology of E is relevant, and not a particular choice of norm. Let Ω be a set and E be a fixed Banachble space of bounded maps on Ω satisfying the left composition property. Given manifolds M, N, we will denote by C∞(M,N) the set of all smooth maps from M to N. For an arbitrary manifold M, we set: M(Ω,M) = f ∈ M Ω : α ◦ f ∈ E, for all α ∈ C∞(M, IR) . We endow M(Ω,M) with the topology induced by the left composition maps: LC(α): M(Ω,M) −→ E, where α runs over the set C∞(M, IR). More explicitly, M(Ω,M) has the coarsest topol- ogy for which LC(α) is continuous for every α ∈ C∞(M, IR). Thus, if ρ is an M(Ω,M)- valued map defined on an arbitrary topological space, then ρ is continuous if and only if LC(α) ◦ ρ is an E-valued continuous map for every α ∈ C∞(M, IR). Now we prove that the rule M defined above satisfies axioms (A0)—(A7). First, we have the following: 3.6. Lemma. If E is a Banachble space of bounded maps on a set Ω satisfying the left composition property and if the rule M is defined as above then, for every n ≥ 1, the topological spaces M(Ω, IRn) and En are equal. Proof. The fact that E satisfies the left composition property implies that En is contained in n n n n M(Ω, IR ) and that the inclusion map E ,→ M(Ω, IR ) is continuous. If πi : IR → IR n denotes projection onto the i-th coordinate, then the fact that LC(πi): M(Ω, IR ) → E is (well-defined and) continuous for all i = 1, . . . , n implies that M(Ω, IRn) is contained in n n n E and that the inclusion map M(Ω, IR ) ,→ E is continuous.  3.7. Theorem. If Ω is a set and E is a Banachble space of bounded maps on Ω satisfying the left composition property then the rule M defined above satisfies axioms (A0)—(A7) of Section 2. Proof. Axioms (A0) and (A7) follow from Lemma 3.6. We now prove the other axioms. Proof of axiom (A1). It is easy to see that LC(φ) carries M(Ω,M) to M(Ω,N). For the continuity, it suffices to show that LC(α) ◦ LC(φ) is continuous for all α ∈ C∞(N, IR); but LC(α) ◦ LC(φ) = LC(α ◦ φ) and α ◦ φ ∈ C∞(M, IR). Proof of axiom (A3). If there were some f ∈ M(Ω,M) with non relatively compact image, we could find a smooth map α : M → IR which is unbounded on Im(f) (for instance, there exists a smooth proper map α : M → IR). But this contradicts the fact that α ◦ f ∈ E and E ⊂ B(Ω, IR). Proof of axiom (A4). Choose f ∈ M(Ω,M) with Im(f) ⊂ U and λ ∈ C∞(M, IR) with λ ≡ 1 on Im(f) and supp(λ) ⊂ U. Given α ∈ C∞(U, IR), then α = λα extends to a smooth map on M that vanishes outside U; moreover, α ◦ f = α ◦ f ∈ E. This proves that f ∈ M(Ω,U).

3Obviously well-defined means that f ∈ En implies φ ◦ f ∈ E. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 15

Proof of axiom (A5). Given f ∈ M(Ω,U), choose λ ∈ C∞(M, IR) with λ ≡ 1 on Im(f) and supp(λ) ⊂ U. If P denotes the open subset of B(Ω, IR) consisting of bounded maps u :Ω → IR with infa∈Ω u(a) > 0 then: f ∈ LC(λ)−1(P ∩ E) ⊂ M(Ω,U), and therefore M(Ω,U) is open in M(Ω,M). Now let M(Ω,U)♦ temporarily denote the space M(Ω,U) endowed with the topology induced from M(Ω,M) and let us show that M(Ω,U)♦ equals M(Ω,U) as a topological space. By axiom (A1), the inclusion map of M(Ω,U) in M(Ω,M) is continuous and hence the identity map:

Id : M(Ω,U) −→ M(Ω,U)♦ is continuous. To prove the continuity of Id : M(Ω,U)♦ → M(Ω,U), we show that ∞ LC(α): M(Ω,U)♦ → E is continuous for every α ∈ C (U, IR). Let f ∈ M(Ω,U) be fixed and choose an open subset V ⊂ M with Im(f) ⊂ V ⊂ V ⊂ U. Choose a map α ∈ C∞(M, IR) that equals α on V . Then the map LC(α) is continuous on M(Ω,M) and hence on M(Ω,U)♦; moreover, LC(α) agrees with LC(α) on M(Ω,V ), which is an open neighborhood of f in M(Ω,M). This proves the continuity of LC(α) on M(Ω,U)♦. −1 Proof of axiom (A6). Let a ∈ Ω be fixed. We show that evala (U) is open in M(Ω,M) −1 for every open set U ⊂ M. Choose f ∈ evala (U); then f(a) ∈ U. Now consider the open subset Pa ⊂ B(Ω, IR) of bounded maps u :Ω → IR with u(a) > 0 and choose α ∈ C∞(M, IR) with αf(a)
= 1 and supp(α) ⊂ U; we have: −1 −1 f ∈ LC(α) (Pa ∩ E) ⊂ evala (U), −1 which proves that evala (U) is open. Proof of axiom (A2). In order to prove that (2.1) is surjective and that its inverse is contin- ∞ uous, we have to show that for every α ∈ C (M1 × M2, IR), the map:

(3.1) M(Ω,M1) × M(Ω,M2) 3 (f, g) 7−→ α ◦ (f, g) ∞ takes values in E and is continuous. Denote by S ⊂ C (M1 × M2, IR) the set of those α for which (3.1) takes values in E and is continuous. It is obvious that S is a subspace ∞ of C (M1 × M2, IR). In fact, S is a subalgebra (under pointwise multiplication) of ∞ 2 C (M1 × M2, IR), because the map LC(m): E → E of left composition with the multiplication map m : IR × IR → IR is continuous. It is also obvious that S contains those α that are independent of one of the two variables. Denote by ni the dimension of ∞ Mi, i = 1, 2. For the rest of the proof, we will say that a map α ∈ C (M1 × M2, IR) has small support if there exists closed sets Fi ⊂ Mi, open sets Ui ⊂ Mi and diffeomorphisms ni ϕi : Ui → IR with Fi ⊂ Ui, i = 1, 2 and supp(α) ⊂ F1 × F2. We claim that S contains ∞ all α ∈ C (M1 × M2, IR) with small support. Namely, for i = 1, 2, choose an open ∞ ni subset Vi ⊂ Mi with Fi ⊂ Vi ⊂ Vi ⊂ Ui, a map ϕi ∈ C (Mi, IR ) that equals ϕi on Vi ∞ and a map λi ∈ C (Mi, IR) with λi ≡ 1 on Fi and supp(λi) ⊂ Vi. Set:

−1 ∞ n1 n2 ∞ αe = α◦(ϕ1 ×ϕ2) ∈ C (IR ×IR , IR), α = αe◦(ϕ1 ×ϕ2) ∈ C (M1 ×M2, IR). It is easy to see that α(x, y) = λ1(x)λ2(y)α(x, y), for all x ∈ M1, y ∈ M2. Thus, to prove that α ∈ S, it suffices to show that α is in S. But this follows by observing that the
map (f, g) 7→ α ◦ (f, g) is equal to the composite LC(αe) ◦ LC(ϕ1) × LC(ϕ2) . ∞ We will now conclude the proof by showing that every α ∈ C (M1 × M2, IR) is in S. Let f0 ∈ M(Ω,M1) and g0 ∈ M(Ω,M2) be fixed. Since Im(f0) and Im(g0) are relatively compact, a simple argument using a partition of unity shows that we can find ∞ Pr maps α1, . . . , αr ∈ C (M1 × M2, IR) with small support such that α equals j=1 αj BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 16 on an open subset of the form U1 × U2 ⊂ M1 × M2 containing Im(f0) × Im(g0). Then Pr j=1 αj is in S and thus the map (f, g) 7→ α ◦ (f, g) is E-valued and continuous on the neighborhood M(Ω,U1) × M(Ω,U2) of (f0, g0). Since f0 and g0 are arbitrary, we have α ∈ S. This concludes the proof.  Let us now give some examples of Banachble spaces of bounded maps satisfying the left composition property. 3.8. Example. Let Ω be a metric space (or, more generally, a uniform space). The space 0 E = Cbu(Ω, IR) of uniformly continuous bounded maps f :Ω → IR endowed with the sup norm is a Banach space of bounded maps. It is easy to see that E has the left composition property. 3.9. Example. Let (Ω, d) be a metric space, α ∈ ]0, 1] and let E = C0,α(Ω, IR) be the space of all α-Holderian¨ bounded maps f :Ω → IR. We define a norm on E by:

f(a) − f(b) kfk = kfksup + sup α . a,b∈Ω d(a, b) a6=b Then E is a Banach space of bounded maps with the left composition property. 3.10. Example. Let Ω be an open subset of IRm, α ∈ ]0, 1], k ≥ 1 and let E = Ck,α(Ω, IR) be the space of all maps f :Ω → IR of class Ck such that f and its partial derivatives up to order k are bounded and α-Holderian.¨ We define a norm on E by:

X X ∂λf(a) − ∂λf(b) kfk = k∂λfksup + sup α , a,b∈Ω ka − bk |λ|≤k |λ|≤k a6=b m where λ = (λ1, . . . , λm) ∈ IN denotes a multi-index. Then E is a Banach space of bounded maps with the left composition property. 3.11. Example. Let Ω be a manifold endowed with a connection on the tangent bundle T Ω and with a Riemannian metric. If f :Ω → IR is a map of class Ck then for i = 1, . . . , k, i N ∗ the i-th covariant derivative ∇ f of f is a section of the tensor bundle i T Ω, on which there is a natural Riemannian structure induced by the Riemannian metric of Ω. We denote k k i by E = Cb (Ω, IR) the space of all bounded maps f :Ω → IR of class C for which k∇ fk is bounded for all i = 1, . . . , k. We define a norm on E by: k X i kfk = kfksup + ∇ f sup. i=1 Then E is a Banach space of bounded maps with the left composition property.

3.12. Example. Let Ω be a set and let E1, E2 be Banachble spaces of bounded maps in Ω with the left composition property. Set E = E1 ∩ E2 and consider E endowed with the topology induced by the two inclusion maps E → Ei, i = 1, 2. If k · ki is a norm for Ei, i = 1, 2, then k · k = k · k1|E + k · k2|E is a norm for E. Obviously E is a Banach space with continuous inclusion in both E1 and E2 (and hence in B(Ω, IR)). Moreover, it is easy to see that E also has the left composition property. 3.13. Remark. If E is a Banachble space of bounded maps on a set Ω satisfying the left composition property then there exists a unique way of defining the topological spaces M(Ω,M) so that axioms (A0)—(A7) are satisfied and M(Ω, IR) = E. Namely, if M is a BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 17 manifold and φ : M → IRn is an embedding into Euclidean space then, by Lemma 2.2, LC(φ) has to be a homeomorphism onto the space: f ∈ M(Ω, IRn): Im(f) ⊂ Im(φ) , with the topology induced from M(Ω, IRn). But axiom (A2) implies that M(Ω, IRn) = En. 3.14. Remark. If M is a rule satisfying axioms (A0)—(A7) then E = M(Ω, IR) must be a Banachble space contained in B(Ω, IR), by axioms (A3) and (A7). By axiom (A6), the inclusion map E → B(Ω, IR) has closed graph and it is therefore continuous. Moreover, axioms (A1) and (A2) imply that E must have the left composition property. Thus (keeping in mind Lemma 3.6, Theorem 3.7 and Remark 3.13), M 7→ E = M(Ω, IR) gives a one to one correspondence between rules M satisfying axioms (A0)—(A7) and Banachble spaces E of bounded maps satisfying the left composition property. 3.1. When constant maps are missing. Let Ω be a set and let E be a Banachble space of bounded maps on Ω. If E satisfies the left composition property, then E must contain the constant maps; namely, if φ : IR → IR is a constant map then LC(φ) must take values in E. Thus, if a Banachble space of bounded maps E does not contain the constants, one cannot hope that E could have the left composition property. But there are important natural function spaces that do not contain the constants. We will deal with this problem now. 3.15. Definition. Let Ω be a set and let E be a Banachble space of bounded maps on Ω. We say that E has the left composition property of type zero if for every n ≥ 1 and every smooth map φ : IRn → IR with φ(0) = 0 the left composition map LC(φ): En → E is well-defined and continuous. We have a simple lemma. 3.16. Lemma. Let E be a Banachble space of bounded maps on a set Ω satisfying the left composition property of type zero. If E contains the constant maps then E satisfies the left composition property. Proof. Given φ ∈ C∞(IRn, IR), set ψ = φ − φ(0) ∈ C∞(IRn, IR) and observe that n LC(φ) = LC(ψ) + c, where c : E → E denotes the constant map equal to φ(0) ∈ E.  If a Banachble space of bounded maps E does not contain the constants, one may hope that, by adding the constants to E, we may obtain a space that satisfies the left composition property. With this in mind, we give the following: 3.17. Definition. Let Ω be a set and let E be a Banachble space of bounded maps on Ω. We say that E has the parametric left composition property of type zero if for every s, n ≥ 1 and every smooth map φ : IRn × IRs → IR such that φ(0, c) = 0, for all c ∈ IRs, the map: n s E × IR 3 (f, c) 7−→ φc ◦ f ∈ E, n s is well-defined and continuous, where φc = φ(·, c): IR → IR, for all c ∈ IR . Obviously the parametric left composition property of type zero implies the left com- position property of type zero. If Ω is a set we denote by Ct(Ω) the unidimensional space of constant maps f :Ω → IR; we identify Ct(Ω)n with the space of constant maps f :Ω → IRn. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 18

3.18. Lemma. Let E0 be a Banachble space of bounded maps on a set Ω satisfying the parametric left composition property of type zero. If Ct(Ω) 6⊂ E0 (or, equivalently, if Ct(Ω)∩E0 = {0}) then E = E0 ⊕Ct(Ω) is a Banachble space of bounded maps satisfying the left composition property. Proof. Obviously E has continuous inclusion in B(Ω, IR). Let φ : IRn → IR be a smooth n n n ∼ n map. If we write the elements of E as pairs (f, c) with f ∈ E0 , c ∈ IR = Ct(Ω) , then the left composition map LC(φ) is given by: n
LC(φ): E 3 (f, c) 7−→ ψc ◦ f, φ(c) ∈ E, n n where ψ : IR × IR → IR is given by ψ(x, c) = φ(x + c) − φ(c) and ψc = ψ(·, c).  We now use the trick of “adding the constants” explained above to deal with some interesting examples. Let (Ω, A, µ) be a measure space, i.e., Ω is a set, A is a σ-algebra of subsets of Ω and µ : A → [0, +∞] is a countably additive measure. Choose a real number p ∈ [1, +∞[ and p consider the space Lb(Ω, IR) consisting of bounded measurable maps f :Ω → IR such that: Z 1  p  p kfkp = |f| dµ < +∞. Ω p It is easy to see that Lb(Ω, IR) endowed with the norm:

kfk = kfksup + kfkp 4 p is a Banach space . If µ(Ω) < +∞ then Lb(Ω, IR) coincides with the closed subspace Mb(Ω, IR) of B(Ω, IR) (see Example 3.2) and the norm above is equivalent to the sup norm. We will therefore focuss on the case µ(Ω) = +∞; observe that in this case p Lb(Ω, IR) does not contain the constant maps. p 3.19. Lemma. The space E = Lb(Ω, IR) is a Banachble space of bounded maps satisfying the parametric left composition property of type zero. Proof. Obviously E has continuous inclusion in B(Ω, IR). Let φ : IRn × IRs → IR be a smooth map with φ(0, c) = 0 for all c ∈ IRs; let us show that the map: n s (3.2) E × IR 3 (f, c) 7−→ φc ◦ f ∈ E s is well-defined and continuous, where φc = φ(·, c). For fixed c ∈ IR , the map φc is n Lipschitz on every bounded subset of IR and, since φc(0) = 0, we have:
φc f(a) ≤ C f(a) , a ∈ Ω, n for some constant C ≥ 0 dependent on f ∈ E . This shows that kφc ◦ fkp < +∞ and obviously φc ◦ f is measurable and bounded. Thus (3.2) is well-defined. To prove the continuity of (3.2), let f ∈ En and c ∈ IRs be fixed and choose a compact convex set K ⊂ IRn containing the origin and with Im(f) contained in the interior of K. Then the set of those g ∈ En with Im(g) ⊂ K is a neighborhood of f in En; for such g and any d ∈ IRs, a ∈ Ω, we compute:





φ f(a), c − φ g(a), d ≤ φ f(a), c − φ g(a), c + φ g(a), c − φ g(a), d

≤ C f(a) − g(a) + sup dφc(x) − dφd(x) g(a) , x∈K where C is a Lipschitz constant for φc on K. From the inequalities above it is easy to see s n that φd ◦ g tends to φc ◦ f in E as d tends to c in IR and g tends to f in E . 

4We do not identify maps that are equal almost everywhere! BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 19

p 3.20. Corollary. If µ(Ω) = +∞ then the space E = Lb(Ω, IR) ⊕ Ct(Ω) is a Banachble space of bounded maps satisfying the left composition property. Proof. Follows from Lemmas 3.18 and 3.19.  If Ω is an open subset of IRm and p ∈ [1, +∞[ is a real number, we denote as usual by Lp(Ω, IR) the space of (equivalence classes of almost everywhere equal) measurable maps k,p f :Ω → IR with kfkp < +∞. For every integer k ≥ 1, we denote by W (Ω, IR) the p p Sobolev space of maps f ∈ L (Ω, IR) such that ∂λf ∈ L (Ω, IR) for every multi-index m λ = (λ1, . . . , λm) ∈ IN with |λ| ≤ k; the partial derivative ∂λ is understood in the distributional sense. As usual, the norm on W k,p(Ω, IR) is defined by: X kfk = k∂λfkp |λ|≤k and k · k turns W k,p(Ω, IR) into a Banach space. For all the results that we will use about Sobolev spaces we refer to [1]. If Ω is a bounded domain with smooth boundary then the theory of Banach manifolds modelled on the Sobolev spaces W k,p(Ω, IR) is well- m k,p 0 known. For instance, if k > p then W (Ω, IR) has continuous inclusion on Cb(Ω, IR) and it satisfies the left composition property (see [4, Lemma 9.9]). We will therefore focuss only on the case that Ω is unbounded and, for simplicity, we take Ω = IRm. Ob- viously, W k,p(IRm, IR) doesn’t contain the constant maps because Lp(IRm, IR) doesn’t. m k,p m 0 m If k > p , the space W (IR , IR) has continuous inclusion in Cb(IR , IR). Since the ∞ m m space Cc (IR , IR) of smooth real valued maps with compact support on IR is dense in W k,p(IRm, IR), it follows that the elements of W k,p(IRm, IR) tend to zero at infinity. We are now going to look for conditions under which the space W k,p(IRm, IR) satisfies the parametric left composition property of type zero. Unfortunately, contrary to the case m 5 of bounded Ω, the condition k > p alone does not imply such property . Not all is lost, though, as it is shown in the following: 3.21. Lemma. If m < p and k ≥ 1 then the Sobolev space W k,p(IRm, IR) is a Banachble space of bounded maps in IRm satisfying the parametric left composition property of type zero. m k,p m Proof. Since k ≥ 1 > p , the space E = W (IR , IR) has continuous inclusion in 0 m m n s Cb(IR , IR) and thus in B(IR , IR). Let φ : IR × IR → IR be a smooth map with s n s φ(0, c) = 0 for all c ∈ IR ; let us show that the map E × IR 3 (f, c) 7→ φc ◦ f ∈ E is ∞ m n n well-defined and continuous, where φc = φ(·, c). Since Cc (IR , IR ) is dense in E , by standard arguments, it suffices to show that the map: ∞ m n s p m Cc (IR , IR ) × IR 3 (f, c) 7−→ ∂λ(φc ◦ f) ∈ L (IR , IR), n s m has a continuous extension to E × IR for every multi-index λ = (λ1, . . . , λm) ∈ IN with |λ| ≤ k. The case λ = 0 (i.e., ∂λ(φc ◦ f) = φc ◦ f) follows from Lemma 3.19, p m m observing that E has continuous inclusion in Lb(IR , IR). If λ 6= 0 then, for a ∈ IR , ∂λ(φc ◦ f)(a) is a linear combination of terms of the form:
(∂βφc) f(a) (∂γ1 fi1 )(a)(∂γ2 fi2 )(a) ··· (∂γr fir )(a), n m where f = (f1, . . . , fn), i1, . . . , ir ∈ {1, . . . , n} and β ∈ IN , γ1,..., γr ∈ IN are Pr n nonzero multi-indices with |β| ≤ |λ| and i=1 |γi| = |λ|. Since E has continuous

5The difficulty of the case Ω = IRm lies on the fact that the continuous embedding W k,p(IRm, IR) ,→ q m 1 1 k 1 1 k L (IR , IR) holds only for q = p − m and not in general for q ≥ p − m , as it does in the case of bounded domains. Thus the proof of [4, Theorem 9.4] doesn’t work if the domain is unbounded. BANACH MANIFOLD STRUCTURE FOR GENERAL SETS OF MAPS 20

0 m n 0 m inclusion in Cb(IR , IR ), we have that (f, c) 7→ (∂βφc)◦f defines a Cb(IR , IR)-valued continuous map on En × IRs. Since the multiplication map: 0 m p m p m (3.3) Cb(IR , IR) × L (IR , IR) 3 (g1, g2) 7−→ g1g2 ∈ L (IR , IR) Qr is continuous, to complete the proof it suffices to show that f 7→ u=1 ∂γu fiu defines a p m n continuous L (IR , IR)-valued map on E . If r = 1 then |γ1| ≤ k and f 7→ ∂γ1 fi1 is p m n a continuous L (IR , IR)-valued map on E . Assume now that r > 1. Then |γu| < k 1,p m for all u = 1, . . . , r and then f 7→ ∂γu fiu defines a continuous W (IR , IR)-valued map on En. The hypothesis m < p implies that W 1,p(IRm, IR) has continuous inclusion 0 m 0 m in Cb(IR , IR) and, since the multiplication map of Cb(IR , IR) is continuous, it follows that the map: r−1 n Y 0 m E 3 f 7−→ ∂γu fiu ∈ Cb(IR , IR) u=1 n p m is continuous. Finally, since E 3 f 7→ ∂γr fir ∈ L (IR , IR) is continuous, the conclu- sion follows from the continuity of (3.3).  3.22. Corollary. Under the hypothesis of Lemma 3.21, W k,p(IRm, IR) ⊕ Ct(IRm) is a Banachble space of bounded maps in IRm satisfying the left composition property.

Proof. Follows from Lemmas 3.18 and 3.21.  Observe that the important case Hk(IR, IR) = W k,2(IR, IR) satisfies the hypothesis of Lemma 3.21.

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DEPARTAMENTO DE MATEMATICA´ , UNIVERSIDADEDE SAO˜ PAULO,BRAZIL E-mail address: [email protected] URL: http://www.ime.usp.br/˜tausk