Manifolds of mappings on Cartesian products

Given smooth manifolds M1,…,Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1,\ldots , M_n$$\end{document} (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and α∈(N0∪{∞})n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$\end{document}, we consider the set Cα(M1×⋯×Mn,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\alpha (M_1\times \cdots \times M_n,N)$$\end{document} of all mappings f:M1×⋯×Mn→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:M_1\times \cdots \times M_n\rightarrow N$$\end{document} which are Cα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\alpha $$\end{document} in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders ≤αj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le \alpha _j$$\end{document} in the jth variable for j∈{1,…,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \{1,\ldots , n\}$$\end{document}, in local charts. We show that Cα(M1×⋯×Mn,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\alpha (M_1\times \cdots \times M_n,N)$$\end{document} admits a canonical smooth manifold structure whenever each Mj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_j$$\end{document} is compact and N admits a local addition. The case of non-compact domains is also considered.


Introduction and statement of the results
As known from classical work by Eells [9], the set C (M, N ) of all C -maps f : M → N can be given a smooth Banach manifold structure for each ∈ N 0 , compact smooth manifold M and σ -compact finite-dimensional smooth manifold N . More generally, C (M, N ) is a smooth manifold for each ∈ N 0 ∪{∞}, locally compact, paracompact smooth manifold M with rough boundary in the sense of [15] (this includes finite-dimensional manifolds with boundary, and manifolds with corners as in [7,8,21]) and each smooth manifold N modeled on locally convex spaces such that N admits a local addition (a concept recalled in Defini-H. Glöckner was supported by German Academic Exchange Service, DAAD Grant 57568548. B Alexander Schmeding alexander.schmeding@nord.no Helge Glöckner glockner@math.uni-paderborn.de tion 5.6); see [4,14,16,21,22,25] for discussions in different levels of generality, and [20]  is C k if and only if the corresponding map of two variables, is C k, in the sense of [3], i.e., a continuous map which in local charts admits up to directional derivatives in the second variable, followed by up to k directional derivatives in the first variable, with continuous dependence on point and directions (see 2.11 and 2.12 for details). We thus obtain a bijection As our first result, for compact L we construct a smooth manifold structure on C k, (L×M, N ) which turns into a C ∞ -diffeomorphism. More generally, analogous to the n = 2 case of C k, -maps, we consider N -valued C α -maps on an n-fold product M 1 × · · · × M n of smooth manifolds for any n ∈ N and α = (α 1 , . . . , α n ) ∈ (N 0 ∪ {∞}) n . With terminology explained presently, we get: Theorem 1.1 Given α = (α 1 , . . . , α n ) ∈ (N 0 ∪ {∞}) n , let M j for j ∈ {1, . . . , n} be a compact smooth manifold with rough boundary. Let N be a smooth manifold modeled on locally convex spaces such that N can be covered by local additions. Then, the set C α (M 1 × · · · × M n , N ) admits a smooth manifold structure which is canonical. The following hold for this canonical manifold structure: (a) C α (M 1 × · · · × M n , N ) can be covered by local additions. If N admits a local addition, then also C α (M 1 × · · · × M n , N ) admits a local addition. (b) Given m ∈ N and β = (β 1 , . . . , β m ) ∈ (N 0 ∪ {∞}) m , let L j be a compact smooth manifold with rough boundary for j ∈ {1, . . . , m}. Then, canonical smooth manifold structures turn the bijection C β (L 1 × · · · × L m , C α (M 1 × · · · × M n , N )) → C β,α (L 1 × · · · × L m × M 1 × · · · × M n , N ) taking g to g ∧ into a C ∞ -diffeomorphism.
The following terminology was used: We say that a smooth manifold N can be covered by local additions if N is the union of an upward directed family (N j ) j∈J of open submanifolds N j which admit a local addition. For instance, any (not necessarily paracompact) finite-dimensional smooth manifold has this property, e.g., the long line. We also used canonical manifold structures.
Canonical manifold structures are essentially unique whenever they exist, and so are precanonical ones (see Lemma 4.3 (b) for details). We address two further topics for not necessarily compact domains: (i) We formulate criteria ensuring that C α (M 1 × · · · × M n , G) admits a canonical smooth manifold structure (making the latter a Lie group), for a Lie group G modeled on a locally convex space; (ii) Manifold structures on C α (M 1 × · · · × M n , N ) which are modeled on certain spaces of compactly supported T N-valued functions, in the spirit of [21].
To discuss (i), we use a generalization of the regularity concept introduced by John Milnor [22] (the case r = ∞). If G is a Lie group modeled on a locally convex space, with neutral element e, we write λ g : G → G, x → gx for left translation with g ∈ G and consider the smooth left action of G on its tangent bundle. We write g := T e G for the Lie algebra of G. Let r ∈ N 0 ∪ {∞}. The Lie group G is called C r -semiregular if, for each C r -curve γ : [0, 1] → g, the initial value problemη is smooth, then G is called C r -regular (cf. [12]). If s ≤ r and G is C s -regular, then G is C r -regular (see [12]). We show: Theorem 1.3 Let G be a C r -regular Lie group modeled on a locally convex space with r ∈ N 0 ∪ {∞}. For some n ∈ N, let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary and α ∈ (N 0 ∪ {∞}) n . For each j ∈ {1, . . . , n} such that M j is not compact, assume that α j ≥ r + 1 and M j is 1-dimensional with finitely many connected components. Then, we have: (a) C α (M 1 × · · · × M n , G) admits a canonical smooth manifold structure; (b) The canonical manifold structure from (a) makes C α (M 1 × · · · × M n , G) a C r -regular Lie group.
The Lie algebra of C α (M 1 × · · · × M n , G) can be identified with the topological Lie algebra C α (M 1 × · · · × M n , L(G)) in a standard way (Proposition 6.6). Of course, we are most interested in the case that the non-compact 1-dimensional factors are σ -compact and hence intervals, or finite disjoint unions of such. But we did not need to assume σ -compactness in the theorem, and thus M j with α j ≥ r + 1 might well be a long line, or a long ray. Disregarding the issue of being canonical, the Lie group structure on C ∞ (M 1 × · · · × M n , G) = C α (M 1 × · · · × M n , G) with α 1 := · · · := α n = ∞ was first obtained in [24], for smooth manifolds M j without boundary which are compact or diffeomorphic to R. The Lie group structure for n = 1 was first obtained in [2] for domains diffeomorphic to intervals, together with a sketch for the case n = 2 (assuming additional conditions, e.g., α 1 ≥ r + 3 and α 2 ≥ r + 1 if M 1 = M 2 = R). Our approach differs: While the studies in [24] and [2] assume regularity of G from the start to enforce exponential laws, and build it into a notion of Lie group structures on mapping groups that are "compatible with evaluations," we take canonical and pre-canonical manifold structures as the starting point (independent of regularity) and combine them with regularity or compatibility with evaluations (adapted to C α -maps in Definition 6.2) only when needed. As to topic (b), our constructions show: In the case that n = 1, k = ∞ and M := M 1 is a smooth manifold with corners, we recover the smooth manifold structure on C ∞ (M, N ) discussed by Michor [21]. Using manifold structures on infinite Cartesian products of manifolds making them "fine box products" (a concept recalled in Sect. 7), Theorem 1.4 turns into a corollary to Theorem 1.1.
In the case n = 1, for compact M and ∈ N 0 ∪ {∞}, canonical manifold structures on C (M, N ) as in Theorem 1.1 have already been considered in [4], in a weaker sense (fixing m = 1 in Definition 1.2). Parts of our discussion adapt arguments from [4] to the more difficult case of C α -maps.

Consider locally convex spaces E, F and a map
exist for all j ∈ N 0 such that j ≤ k, x ∈ U and y 1 , . . . , y j ∈ E, and the maps d j f : U × E j → F are continuous, then f is called C k . If U may not be open, but has dense interior U o and is locally convex in the sense that each x ∈ U has a convex neighborhood in U , following The C ∞ -maps are also called smooth.

Remark 2.2
If E = R n and U is relatively open in [0, ∞[ n , then f as above is C k if and only f has a C k -extension to an open set in R n (see [13], cf. [17]).

2.3
Let k ∈ N ∪ {∞}. A manifold with rough boundary modeled on a non-empty set E of locally convex spaces is a Hausdorff topological space M, together with a set A of homeomorphisms ("charts") φ :  [4] for modifications in the general case).

2.4
All manifolds and Lie groups considered in the article are modeled on locally convex spaces which may be infinite-dimensional, unless the contrary is stated. Finite-dimensional manifolds need not be paracompact or σ -compact, unless stated explicitly. As we are interested in manifolds of mappings, consideration of pure manifolds would not be sufficient.

If
U is an open subset of a locally convex space E (or a locally convex subset with dense interior), we identify its tangent bundle T U with U × E, as usual, with bundle projection (x, y) → x. If M is a C k -manifold with rough boundary and f : using the bundle projection π T M : T M → M.

2.6
If G is a Lie group with neutral element e, we write L(G) := T e G (or g) for its tangent space at e, endowed with its natural topological Lie algebra structure. If ψ : G → H is a smooth homomorphism between Lie groups, we let L(ψ) := T e ψ : L(G) → L(H ) be the associated continuous Lie algebra homomorphism.

2.7
If G is a Lie group with Lie algebra g and I a non-degenerate interval with 0 ∈ I , we If G is C r -semiregular and γ ∈ C k (I , g), then there exists a unique η ∈ C 1 (I , g) such that η(0) = e and δ (η) = γ . Moreover, η is C k+1 .

2.9
Let M be a smooth manifold (without boundary) is a locally convex subset of E φ with dense interior.

2.11
Let F and E 1 , . . . , E n be locally convex spaces, U j ⊆ E j be an open subset for j ∈ {1, . . . , n} and f : U → F be a map on U := U 1 ×· · ·×U n . Identifying E := E 1 ×· · ·× E n with E 1 ⊕ · · · ⊕ E n , we can identify each E j with a vector subspace of E, and simply write D y f (x) for a directional derivative with x ∈ U , y ∈ E j (rather than D (0,...,0,y,0,...,0) f (x) with j −1 zeros on the left and n− j zeros on the right-hand side). For y = (y 1 , . . . , y k ) ∈ E k j , abbreviate

2.12
Let M 1 , . . . , M n be C ∞ -manifolds with rough boundary, α ∈ (N 0 ∪ {∞}) n and N be a C k -manifold with k ≥ |α|. We say that a map f : M 1 × · · · × M n → N is C α if, for each x = (x 1 , . . . , x n ) ∈ M 1 × · · · × M n , there are charts φ j : U j → V j for M j around x j for j ∈ {1, . . . , n} and a chart ψ : is C α . The latter then holds for any such charts, by the chain rule for C α -maps (as in [1,Lemma 3.16]).
Let r ∈ N 0 ∪ {∞}, E 1 , . . . , E n and F be locally convex spaces and U j be a locally convex subset of E j with dense interior, for j ∈ {1, . . . , n}. We mention that a map f : U 1 × · · · × U n → F is C r if and only if it is C β for all β ∈ (N 0 ∪ {∞}) n such that |β| ≤ r . More generally, the following is known (as first formulated and proved in the unpublished work [18]):

The compact-open C˛-topology
As a further preliminary, we introduce a topology on C α (M 1 × · · · × M n , N ) which parallels the familiar compact-open C k -topology on C k (M, N ). Basic properties are recorded, with proofs in "Appendix A".

Definition 3.4
The compact-open C α -topology on C α (M, N ) is the initial topology with respect to the mappings Pushforwards and pullbacks are continuous.

Lemma 3.5 Using compact-open C α -topologies, we have:
(a) If L is a smooth manifold with rough boundary and g : N → L a smooth map, then the following map is continuous: (b) Let L j be a smooth manifold with rough boundary for j ∈ {1, . . . , n} and g j : L j → M j be a smooth map. Abbreviate L := L 1 × · · · × L n and g := g 1 × · · · × g n . Then, the following map is continuous:

Remark 3.6
If L j is a full submanifold of M j for j ∈ {1, . . . , m}, then the inclusion map g j : L j → M j , x → x is smooth. By Lemma 3.5 (b), the map Lemma 3.7 Let (K i ) i∈I be a family of subsets K i ⊆ M whose interiors K o i cover M, such that K i = K i,1 × · · · × K i,n for certain full submanifolds K i, j ⊆ M j for j ∈ {1, . . . , n}. Then, the compact-open C α -topology on C α (M, N ) is initial with respect to the restriction is an isomorphism of topological vector spaces.
Using the multiplication R × T N → T N, (t, v) → tv with scalars, we have: In [1], exponential laws were provided for function spaces on products of pure manifolds. The one we need remains valid for manifolds which need not be pure: The next lemma describes the C α -topology on C α (U , F) more explicitly. It will not be used here. The proof, which parallels the C k -case in [15, Lemma 4.1.12], can be found in the preprint version of this article, arXiv:2109.01804. Lemma 3.14 Let E j be a locally convex space for j ∈ {1, . . . , n} and U j ⊆ E j be a locally convex subset with dense interior. Let F be a locally convex space, α ∈ (N 0 ∪ {∞}) n , and U := U 1 × · · · × U n . Then, the compact-open C α -topology on C α (U , F) is initial with respect to the maps

(Pre-)Canonical manifold structures
In this section, we establish basic properties of canonical manifolds of mappings, and precanonical ones. We begin with examples. (a) Let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary and E a locally convex space. Then, C α (M 1 × · · · × M n , E) is a canonical manifold due to Lemma 3.13. The same holds for C α (M 1 × · · · × M n , N ) if N is a smooth manifold diffeomorphic to E, endowed with the C ∞ -manifold structure making ϕ * : Familiar examples of mapping groups turn out to be canonical, notably loop groups C k (S 1 , G) for G a Lie group, and certain Lie groups of the form C k (R, G) discussed in [2,24]. We extend these constructions in Sect. 6.
We will now establish general properties of canonical manifolds.

Conventions
We denote by α, β multiindices in (N 0 ∪ {∞}) n for some n ∈ N. Likewise we will usually adopt the shorthand M:=M 1 × M 2 × · · · × M n where the M i are locally compact manifolds (possibly with rough boundary). If M is the domain of definition of the function space C α (M, N ) we will assume that the number of entries of the multiindex α coincides with the number of factors in the product M.

Lemma 4.3 If C α (M, N ) is endowed with a pre-canonical manifold structure, then the following holds:
(a) The evaluation map ev : Pre-canonical manifold structures are unique in the following sense: If we write C α (M, N ) for C α (M, N ) with another pre-canonical manifold structure, then id : Then, the submanifold structure on C α (M, S) is pre-canonical.
k be a product of smooth manifolds (possibly with rough boundary) modeled on locally convex spaces and f :

Remark 4.4
Note that due to Lemma 4.3 (a), the evaluation on a canonical manifold is a C ∞,α -map whence it is at least continuous. For a C k -manifold M which is C k -regular 1 and a locally convex space E = {0}, it is well known that for the compact-open C k -topology the evaluation ev : C k (M, E) × M → E is continuous if and only if M is locally compact. A similar statement holds for the compact-open C α -topology. Using a chart for N and cutoff functions, we deduce that the evaluation of C α (M, N ) is discontinuous if M fails to be locally compact, provided N is not discrete and M is C |α| -regular; then C α (M, N ) cannot admit a canonical manifold structure.
We now turn to smoothness properties of the composition map.

Corollary 4.6
If C α (M, N ) and C α (M, L) are endowed with pre-canonical manifold structures, then the pushforward f * : .
The chain rule also allows the following result to be deduced.
Proof Due to the chain rule, the pullback g * makes sense.
The key point was the differentiability of the evaluation map together with a suitable chain rule. Thus, by essentially the same proof, one obtains from the chain rule [1, Lemma 3.16] the following statement whose proof we omit.

Proposition 4.9
Assume that all the manifolds of mappings occurring in the following are endowed with pre-canonical manifold structures. Further, we let Then, for σ = (σ 1 , . . . , σ n ) and α = (α 1 , . . . , α n ), the composition map The above discussion shows that composition, pushforward, and pullback maps inherit differentiability and continuity properties. The following variant will be used in the construction process of canonical manifold structures.
is an open subset of C α (K , M) and Proof By compactness of K , the compact-open topology on C(K , M) coincides with the graph topology (see, e.g., [15, Proposition A.6.25]). Thus, is open in C α (K , M). By Lemma 4.3 (a), the evaluation ev : Lemma 3.16] shows that For later use, we record several observations on stability of (pre-)canonical structures under pushforward by diffeomorphisms.

Lemma 4.11 Let N 1 and N 2 be smooth manifolds and
then the smooth manifold structure on C α (M, N 1 ) turning the bijection be endowed with a pre-canonical manifold structure and assume that both C β (L, C α (M, N )) and C β,α (L × M, N ) are smooth manifolds making the bijection and only if f 1 and f 2 are C β . As the manifold structures are (pre-)canonical, this holds if and only if As ψ is a smooth diffeomorphism we deduce from the chain rule that this is the case if and only if f ∧ is of is even canonical, the C αtopology is transported by the diffeomorphism ψ * to the C α -topology on C α (M, N 1 ). Hence, the manifold C α (M, N 1 ) is also canonical in this case.
(c) By construction, a map f : N ) are smooth manifolds such that the bijection N ) is pre-canonical. The converse assertion for C α•σ (M, N ) follows verbatim by replacing φ σ with its inverse. Note that if one of the manifolds is even canonical, it follows directly from the definition of the C α -topology, Definition 3.4, that reordering the factors induces a homeomorphism of the C α -and C α•σ -topology. Hence, we see that one of the manifolds is canonical if and only if the other is so. (b) Note that the inverse of φ * σ is (φ −1 σ ) * whence the situation is symmetric and it suffices to prove that φ * σ (and by an analogous argument also its inverse) is smooth.
) being a C ∞,α -mapping. This follows from Lemma 4.3 (a), the chain rule, and Lemma 2.17. (c) Replacing φ σ with ψ 1 ×· · ·×ψ n , the argument is analogous to (b). If C α (M, N ) is canonical, then the C α -topology pulls back to the C α -topology under the diffeomorphism, by Lemma 3.5.
An exponential law is available for pre-canonical smooth manifold structures.  N )) the smooth manifold structure making a C ∞diffeomorphism, then this structure is pre-canonical by Lemma 4.11 (c). It therefore coincides with the given pre-canonical smooth manifold structure thereon, up to the choice of modeling spaces (Lemma 4.3 (b)).
There is a natural identification of tangent vectors for pre-canonical manifolds, in good cases.
We thus obtain a map Under additional assumptions, one can show that is a diffeomorphism, allowing tangent vectors v ∈ T C α (M, N ) to be identified with (v). We will encounter a setting in which this statement becomes true in the next section (see Theorem 5.14).

Constructions for compact domains
We now construct and study manifolds of C α -mappings on compact domains. The results of this section subsume Theorem 1.1. They generalize constructions for C k, -functions in [4, Appendix A].

5.1
Let N be a smooth manifold, α ∈ (N 0 ∪ {∞}) n and M = M 1 × · · · × M n be a locally compact smooth manifold with rough boundary. If π : E → N is a smooth vector bundle over N and f : M → N is a C α -map, then we define Pointwise operations turn f into a vector space.
Let us prove that f is a locally convex space. To this end, we cover N with open sets (U i ) i∈I on which the restriction Combining continuity of f and local compactness of M we can find families K j of full compact submanifolds of M j with the following properties: The interiors of the sets in K j cover M j . There is a set K ⊆ 1≤ j≤n K j such that for every K = K 1 × · · · × K n ∈ K we have f (K ) ⊆ U i K for some i K ∈ I and the interiors of the submanifolds in K cover M. Hence, we deduce from Lemma 3.7 that the map Restricting to this subset we obtain a topological embedding where the identification exploits Lemma 3.11 and the fact that pushforwards with smooth diffeomorphisms induce homeomorphisms of the C α -topology (see Lemma 3.5). The image of e are precisely the mappings which coincide on the intersections of the compact sets K (see (10) and the explanations there). Hence, we can exploit that point evaluations are continuous on C α (K , E i K ) by [2,Proposition 3.17] to see that the image of e is a closed vector subspace As the space on the right-hand side is locally convex, we deduce that the co-restriction of e onto its image is an isomorphism of locally convex spaces. Thus, f is a locally convex topological vector space.
We will sometimes write f (E) instead of f to emphasize the dependence on the vector bundle E.
The previous setup allows an essential exponential law to be deduced.
Proof With the notation as in 5.1 we identify f via e with a closed subspace of the locally convex space K ∈K C α (K , E i K ) (the identification will be suppressed in the notation). Thus, Lemma 2.14 (a) implies that the map g is C β if and only if the components g K : L → C α (K , E i K ) are C β -maps. By the exponential law [1,Theorem 4.4], the latter holds if and only if the mappings are of class C β,α . Since the interiors of sets K ∈ K cover M, we deduce that this is the case if and only if g ∧ is of class C β,α .

Remark 5.3
If all fibers of E are Fréchet spaces and K is σ -compact and locally compact, then F is a Fréchet space; if all fibers of E are Banach spaces, K is compact, and |α| < ∞, then f is a Banach space. To see this, note that we can choose the family K in 5.1 countable (resp., finite). Suppressing again the identification, is linear and a topological embedding with closed image. If all F j are Fréchet spaces, so is each C α (K j , F j ) (cf., e.g., [15]) and hence also f . If all F j are Banach spaces and |α| as well as J is finite, then each C α (K j , F j ) is a Banach space (cf. loc. cit.) and hence also f .
Observe that the exponential law for f gives this space the defining property of a precanonical manifold (and the only reason we do not call it pre-canonical is that it is only a subset of C α (M, E)). In particular, the proof of Lemma 5.5 Let π 1 : E 1 → N and π 2 : E 2 → N be smooth vector bundles over a smooth manifold N . Let α ∈ (N 0 ∪ {∞}) m and f : M → N be a C α -map on a product M = M 1 × · · · × M n of smooth manifolds with rough boundary. Then, the following holds: for each τ ∈ f (E 1 ) and . Evaluating at points we see that the map f (ψ) is linear; being a restriction of the continuous map C α (M, ψ): 0) and (0, v 2 ), respectively, then is a continuous linear map which is a homeomorphism as it has the continuous map

Construction of the canonical manifold structure
Having constructed spaces of C α -sections as model spaces, we are now in a position to construct the canonical manifold structure on C α (K , M), assuming that M is covered by local additions and K is compact.
we say that the local addition is normalized.
Until Lemma 5.9, we fix the following setting, which allows a canonical manifold structure on C α (K , M) to be constructed.

5.7
We consider a product K = K 1 × K 2 × · · · × K n of compact smooth manifolds with rough boundary, a smooth manifold M which admits a local addition : T M ⊇ U → M, and α ∈ (N 0 ∪ {∞}) n .

Manifold structure on C α (K , M) if M admits a local addition
For f ∈ C α (K , M), let f :={τ ∈ C α (K , T M) : π T M • τ = f } be the locally convex space constructed in 5.1. Then, M), and is a homeomorphism with inverse g → θ −1 •( f , g). By the preceding, if also h ∈ C α (K , M), is a C ∞,α -map (exploiting that the evaluation map ε : f × K → T M is C ∞,α , by Lemma 5.4). Hence, C α (K , M) endowed with the C α -topology has a smooth manifold structure for which each of the maps φ −1 f is a local chart.
We now prove that the manifold structure on C α (K , M) is canonical. Together with Lemma 4.3 (b), this implies that the smooth manifold structure on C α (K , M) constructed in 5.8 is independent of the choice of local addition.

Lemma 5.9
The manifold structure on C α (K , M) constructed in 5.8 is canonical.
Proof We first show that the evaluation map ev : is C β,α by [1, Lemma 3.16], the map φ −1 f • h| W : W → f (and hence also h| W ) is C k , by Lemma 5.2. Proposition 5.10 Let K = K 1 × · · · × K n be a product of compact smooth manifolds with rough boundary and M be a manifold covered by local additions. For every α ∈ (N 0 ∪{∞}) n , the set C α (K , M) can be endowed with a canonical manifold structure.
Proof Let (M j , j ) j∈J be an upward directed family of open submanifolds M j with local additions i whose union coincides with M. As K is compact, we observe that the sets are open in the C α -topology. Following Lemma 5.9, we can endow every C α (K , M j ) with a canonical manifold structure. Now if M j ⊆ M , Lemma 4.3 (c) implies that also the submanifold structure induced by the inclusion C α (K , M j ) ⊆ C α (K , M ) is canonical. Thus, uniqueness of canonical structures, Lemma 4.3 (b), shows that the submanifold structure must coincide with the canonical structure constructed on C α (K , M j ) via 5.8. As C α (K , M) = j∈J C α (K , M j ) and each step of the ascending union is canonical, the same holds for the union.

The tangent bundle of the manifold of mappings
In the rest of this section, we identify the tangent bundle of C α (K , M) as the manifold C α (K , T M) (under the assumption that K is compact and M covered by local additions). To explain the idea, let us have a look at C α (K , T M).

5.11
Consider a smooth manifold M covered by local additions. Then also, T M is covered by local additions, cf. [4, A.11] for the construction. Thus, for K a compact manifold C α (K , M) and C α (K , T M) are canonical manifolds. If we denote by π : T M → M the bundle projection, Corollary 4.6 shows that the pushforward π * : C α (K , T M) → C α (K , M) is smooth. The fibers of π * are the locally convex spaces π −1 * ( f ) = f from 5.1. We deduce that π * : C α (K , T M) → C α (K , M) is a vector bundle (see Theorem 5.14 for a detailed proof).
We will first identify the fibers of the tangent bundle.

5.12
The tangent space T f C α (K , M) is given by equivalence classes [t → c(t)] of C 1curves c : ]−ε, ε[ → C α (K , M) with c(0) = f , where the equivalence relation c 1 ∼ c 2 holds for two such curves if and only ifċ 1 (0) =ċ 2 (0). Since the manifold structure is canonical (Lemma 5.10) we see that c is C 1 if and only if the adjoint map c ∧ : ]−ε, ε[ ×K → N is a C 1,α -map. The exponential law shows that the derivative of c corresponds to the (partial) derivative of c ∧ , i.e., the mapping from (2) restricts to a bijection We wish to glue the bijections on the fibers to identify the tangent manifold as the bundle from 5.11. To this end, we recall a fact from [4, Lemma A.14]:

5.13
If a manifold M admits a local addition, it also admits a normalized local addition.
Hence, we may assume without loss of generality that the local additions in the following are normalized. Moreover, we will write ε x : C α (K , M) → M for the point evaluation in x ∈ K . Then, the tangent bundle of C α (K , M) can be described as follows.  (2), is an isomorphism of smooth vector bundles (over the identity).
If we wish to emphasize the dependence on M, we write M instead of .  M)) and

Proof
takes T f (C α (K , M)) bijectively and linearly onto f . As the manifolds T (C α (K , M)) and C α (K , T M) are the disjoint union of the sets T f (C α (K , M)) and f = π −1 * ({ f }), respectively, we see that is a bijection. If we can show that is a C ∞ -diffeomorphism, π * : C α (K , T M) → C α (K , M) will be a smooth vector bundle over C α (K , M) (like T (C α (K , M))). Finally, will then be an isomorphism of smooth vector bundles over id M . For the proof, we recall some results from the Appendix of Hence, it suffices to prove that the bijective map restricts to a C ∞ -diffeomorphism on these open sets. In other words it suffices to show that M) (as all other mappings in the formula are smooth diffeomorphisms). Now for all (σ, τ ) ∈ O f × f , and thus we can rewrite (T φ f (σ, τ )) as Thus, the desired formula holds and shows that is a C ∞ -diffeomorphism. This concludes the proof.

Remark 5.15
Assume that the local additions : U i → M i covering M are normalized. Then, the proof of Theorem 5.14 shows that (5)).
Using canonical manifold structures, we have: Corollary 5.16 Let K = K 1 ×· · ·× K n be a product of compact smooth manifolds with rough boundary, α ∈ (N 0 ∪ {∞}) n and g : M → N be a C |α|+1 -map between smooth manifolds M and N covered by local additions. Then, the tangent map of the C 1 -map which is continuous linear and corresponds to T f (g * ).
Moreover, the identification of the tangent bundle allows us to lift local additions (cf.

If now M is covered by open submanifolds (M j ) j∈J each admitting a local addition, it suffices to see that C α (K , M j ) is an open submanifold of C α (K , M) which admits a local addition by the above considerations. Thus, C α (K , M) is covered by the open submanifolds (C α (K , M j )) j∈J and as each of those admits a local addition, C α (K , M) is covered by local additions.
Proposition 5.18 Let K = K 1 × · · · × K m and L = L 1 × · · · × L n be products of compact manifolds with rough boundary and M be a manifold covered by local additions. Fix α ∈ (N 0 ∪ {∞}) n , β ∈ (N 0 ∪ {∞}) m . Then, C β,α (L × K , M), C α (K , M) and C β (L, C α (K , M)) admit canonical manifold structures. Using these, the bijection Proof We apply Proposition 5.10 to obtain canonical manifold structures on C α (K , M) and C β,α (L × K , M). By Lemma 5.17, C α (K , M) is covered by local additions. Hence, we may apply Proposition 5.10 again to obtain a canonical manifold structure on C β (L, C α (K , M)). By Proposition 4.13, the bijection C β,α (L × K , M) → C β (L, C α (K , M)) is a diffeomorphism.

Lie groups of Lie group-valued mappings
We now prove Theorem 1.3, starting with observations.
Another concept is useful, with notation as in 2.6.  G) is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations and that C α (N , C β (M, G)), whose definition uses the latter structure, is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations. Endow C α,β (N × M, G) with the smooth manifold structure turning the bijection Then, the preceding smooth manifold structure on C α,β (N × M, G) is pre-canonical and compatible with evaluations.
Proof By Lemma 4.11 (c), the C ∞ -manifold structure on C α,β (N × M, G) is pre-canonical, whence the latter is a Lie group. The C ∞ -diffeomorphism is a homomorphism of groups. Hence, is an isomorphism of topological Lie algebras. Consider the point evaluations ε x : is an isomorphism of topological Lie algebras and so is by the exponential law (Lemma 3.13). Hence, is an isomorphism of topological Lie algebras. Regard v ∈ L(C α,β (N × M, G)) as a geometric tangent vector [γ ] for a smooth curve γ : ( (γ (t))))] We deduce that (L(ε (x,y) )(v)) (x,y)∈N ×M = φ(v) ∈ C α,β (N × M, L(G)). Since φ is an isomorphism of topological Lie algebras, the Lie group structure on C α,β (N × M, G) is compatible with evaluations.

and G be a Lie group. Assume that C α (M, G) is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations. If the Lie group G is C r -regular for some r ∈ N 0 ∪ {∞}, then also the Lie group C α (M, G) is C r -regular.
Proof Consider the smooth evolution map Evol : ) x∈M is an isomorphism of topological Lie algebras. Then also, is an isomorphism of topological Lie algebras. By Example 4.1, the smooth manifold structures on all of the locally convex spaces C r ([0, 1], C α (M, g)), are canonical. By Lemma 3.13, the Lie algebra homomorphism is an isomorphism of topological Lie algebras. Flipping the factors [0, 1] and M (with Lemma 4.12 (b)) and using the exponential law again, we obtain an isomorphism of topological Lie algebras f (t, x). By Theorem 1.1, C r +1 ([0, 1], C α (M, G)) has a canonical smooth manifold structure. Using Lemmas 4.11 (c), 4.12 (a), and 4.11 (c) in turn, we can give C α (M, C r +1 ([0, 1], G)) a pre-canonical smooth manifold structure making the map The structures being precanonical, It remains to show that E is the evolution map of C α (M, G). As the L(ε x ) separate points on and We establish Theorem 1.3 in parallel with the first conclusion of the following proposition, starting with two basic cases: Case 1: The manifolds M 1 , . . . , M n are compact; Case 2: M is 1-dimensional with finitely many connected components. The final assertion is clear: Starting with any canonical structure on C α (M, G) and a chart φ : U φ → V φ → E φ around the constant map e, using left translations (which are C ∞diffeomorphisms) we can create charts around every f ∈ C α (M, G) which are modeled on the given E φ . We can therefore select a subatlas making C α (M, G) a pure smooth manifold.
Since E φ is isomorphic to L (C α (M, G)), which is isomorphic to E := C α (M, L(G)) as a locally convex space (by compatibility with evaluations), we can replace E φ with E. The pure canonical structure modeled on E is unique, since id C α (M,G) is a C ∞ -diffeomorphism for any two canonical structures (cf. Lemma 4.3 (b)). Proof We first assume that M is connected. Let g := L(G) be the Lie algebra of G. If N is a full submanifold of M, we write 1 C k−1 (N , g) ⊆ C k−1 (T N, g) for the locally convex space of g-valued 1-forms on N , of class C k−1 . Using the Maurer-Cartan form can be associated to each f ∈ C k (N , G), called its left logarithmic derivative. Fix x 0 ∈ M. For every σ -compact, connected, full submanifold N ⊆ M such that x 0 ∈ N , there exists a C ∞ -diffeomorphism ψ : I → N for some non-degenerate interval I ⊆ R, such that 0 ∈ I and ψ(0) = x 0 . Then, the diagram is commutative, where ψ * : C k (N , G) → C k (I , G), f → f • ψ and the vertical map θ on the right-hand side, which takes ω to ω •ψ, are bijections. For each ω ∈ 1 C k−1 (N , g), there is a unique f ∈ C k (N , G) such that f (x 0 ) = e and δ N ( f ) = ω: In fact, Lemma 2.8 yields a unique η ∈ C k (I , G) with η(0) = e and δ (η) = θ(ω); then f := (ψ * ) −1 (η) is as required. We set Evol N (ω) := f .
which is a submanifold of 1 C k−1 (M, g). Let K be the set of all connected, compact full submanifolds K ⊆ M such that x 0 ∈ K . By the preceding, δ K (C k (K , G) [16, Theorem 3.5] provides a smooth manifold structure on C k (M, g) which makes it a C r -regular Lie group, is compatible with evaluations, and turns It remains to show that the smooth manifold structure is canonical. To prove the latter, we first note that K is directed under inclusion. In fact, if K 1 , K 2 ∈ K, then K 1 ∪ K 2 is contained in a σ -compact, connected open submanifold N of M (a union of chart domains diffeomorphic to convex subsets of R, around finitely many points in the compact set K 1 ∪ K 2 ). Pick a C ∞ -diffeomorphism ψ : I → N as above. Then, ψ −1 (K 1 ) and ψ −1 (K 2 ) are compact intervals containing 0, whence so is their union. Thus, K 1 ∪ K 2 is a connected, compact full submanifold of N and hence of M. For K , L ∈ K with K ⊆ L, let r K ,L : 1 C k−1 (L, g) → 1 C k−1 (K , g) be the restriction map. As a consequence of Lemma 3.7 and (8), holds as a locally convex space, using the restriction maps r K : 1 C k−1 (M, g) → 1 C k−1 (K , g) as the limit maps. For K ∈ K, let ρ K : C k (M, G) → C k (K , G) be the restriction map; endow C k (K , G) with its canonical smooth manifold structure (as in Lemma 6.7), which is compatible with evaluations (the "ordinary" Lie group structure in [16]). Then, on C k (M, G), using the above Lie group structure making ψ a C ∞diffeomorphism. Let α ∈ (N 0 ∪ {∞}) m , L 1 , . . . , L m be smooth manifolds with rough boundary, L := L 1 × · · · × L m and f : L → C k (M, G) be a map. If f is C α , then also ρ K • f is C α . Since C k (K , G) is canonical, the map The smooth manifold structure on C k (K , G) being canonical, we deduce that ρ K • f is C α . The hypotheses of Lemma 2.15 being satisfied with A := K, C k (M, G) in place of M, M K := C k (K , G), F := 1 C k−1 (M, g), F K := 1 C k−1 (K , g), and N := G, we see that f is C α . The smooth manifold structure on C k (M, G) is therefore pre-canonical. The topology on the projective limit 1 C k−1 (M, g) is initial with respect to the limit maps r K , whence the topology on 1 C k−1 (M, g) × G is initial with respect to the maps r K × id G . Since ψ is a homeomorphism, we deduce that the topology O on the Lie group C k (M, G) is initial with respect to the maps (r K × id G ) • ψ = ψ K • ρ K . Since ψ K is a homeomorphism, O is initial just as well with respect to the family (ρ K ) K ∈K . But also the compact-open C k -topology T on C k (M, G) is initial with respect to this family of maps (see Lemma 3.7), whence O = T and C k (M, G) is canonical. If M has finitely many components M 1 , . . . , M n , we give C k (M, G) the smooth manifold structure turning the bijection into a C ∞ -diffeomorphism. Let ρ j be its jth component. Since ρ is a homeomorphism for the compact-open C k -topologies (cf. Lemma 3.7) and an isomorphism of groups, the preceding smooth manifold structure makes C k (M, G) a Lie group and is compatible with the compact-open C k -topology. As each of the Lie groups C k (M j , G) is C r -regular, also their direct product (and thus C k (M, G)) is C r -regular. Since ρ = (ρ j ) n j=1 is an isomorphism of Lie groups, is an isomorphism of topological Lie algebras. For x ∈ M j , the point evaluation ε x : C k (M, G) → G is smooth, as the point evaluationε x : C k (M j , G) → G is smooth and ε x =ε x • ρ j . We know that φ j (v) := (L(ε x )(v)) x∈M j ∈ C k (M j , g) for all v ∈ L(C k (M j , G)) and that φ j : L(C k (M j , G)) → C k (M j , g) is an isomorphism of topological Lie algebras. For each v ∈ L(C k (M, G)), we have (M, g). Let us show that the Lie algebra homomorphism φ : L(C k (M, G)) → C k (M, g) is a homeomorphism. Lemma 3.7 entails that the map is a homeomorphism. By the preceding, r • φ = (φ 1 × · · · × φ n ) • (L(ρ j )) n j=1 is a homeomorphism, whence so is φ. Thus, the Lie group structure on C k (M, G) is compatible with evaluations. If α, L = L 1 × · · · × L m and f : L → C k (M, G) are as above and f is C α , then f ∧ is C α,k by the above argument. If, conversely, f ∧ is C α,k , then f ∧ | L×M j is C α,k , whence ( f ∧ | L×M j ) ∨ = ρ j • f is C α for all j ∈ {1, . . . , n}. As a consequence, ρ • f is C α and thus also f . We have shown that the smooth manifold structure on C k (M, G) is precanonical and hence canonical, as compatibility with the compact-open C k -topology was already established.
Another lemma is useful. N 1 , . . . , N m and M 1 , . . . , M n be locally compact smooth manifolds with rough boundary, α ∈ (N 0 ∪ {∞}) m , β ∈ (N 0 ∪ {∞}) m , and G be a Lie group. Abbreviate N := N 1 × · · · × N m and M := M 1 × · · · × M n . Assume that C β (M, G) has a pre-canonical smooth manifold structure, using which C α (N , C β (M, G)) has a canonical smooth manifold structure. Endow C α,β (N × M, G) with the pre-canonical smooth manifold structure turning (N , C β (M, G)), f → f ∨ into a C ∞ -diffeomorphism. Assume that there exists a family (K i ) i∈I of compact full submanifolds K i of N whose interiors cover N , with the following properties:

Lemma 6.9 Let
(a) For each i ∈ I , we have K i = K i,1 × · · · × K i,m with certain compact full submanifolds K i, ⊆ N ; and C α (K i , G)) admits a canonical smooth manifold structure for each i ∈ I , using the canonical smooth manifold structure on C α (K i , G) provided by Theorem 1.1.
Then, the pre-canonical manifold structure on C α,β (N × M, G) is canonical.
Proof Let O be the topology on C α,β (N × M, G), equipped with its pre-canonical smooth manifold structure. Using Theorem 1.1, for i ∈ I we endow C α (K i , C β (M, G)) with a canonical smooth manifold structure; the underlying topology is the compact-open C α -topology. The given smooth manifold structure on C α (N , C β (M, G)) being canonical, its underlying topology is the compact-open C α -topology, which is initial with respect to the restriction maps for i ∈ I . We have bijections using in turn the exponential law (in the form (1)), a flip in the factors (cf. Lemma 4.12 (a)), and again the exponential law. If, step by step, we transport the smooth manifold structure from the left to the right, we obtain a pre-canonical smooth manifold structure in each step (see Lemmas 4.11 (c) and 4.12 (a)). As pre-canonical structures are unique, the pre-canonical structure obtained on C β (M, C α (K i , G)) must coincide with the canonical structure which exists by hypothesis. Hence, using this canonical structure, the map Let L k be the set of compact full submanifolds of M k for k ∈ {1, . . . , n}. Write L 1 × · · · × L n =: J . If j ∈ J , then j = (L j,1 , . . . , L j,n ) with certain compact full submanifolds L j,k ⊆ M k ; we define L j := L j,1 × · · · × L j,n . By Lemma 3.7, the topology on C β (M, C α (K i , G)) is initial with respect to the restriction maps using the compact-open C α -topology on the range which underlies the canonical smooth manifold structure given by Theorem 1.1. Let i, j be the composition of the bijections As each of the domains and ranges admits a canonical smooth manifold structure (by Theorem 1.1), all of the maps have to be homeomorphisms (see Proposition 4.13 and Lemma 4.12 (b)). Thus, i, j is a homeomorphism. By transitivity of initial topologies, O is initial with respect to the mappings which are the restriction maps C α,β (N × M, G) → C α,β (K i × L j , G). Also the compactopen C α,β -topology on C α,β (N × M, G) is initial with respect to the maps ρ i, j , and hence coincides with O. The given pre-canonical smooth manifold structure on C α,β (N × M, G) therefore is canonical.
is an isomorphism of topological vector spaces, being a composition of such.

Proof of Theorem 1.3 and Proposition 6.6.
Step 1. We first assume that M j is 1-dimensional with finitely many components for all j ∈ {1, . . . , n}, and prove the assertions by induction on n. The case n = 1 was treated in Lemma 6.8. We may therefore assume that n ≥ 2 and assume that the conclusions hold for n −1 factors. We abbreviate k := α 1 , β := (α 2 , . . . , α n ), and L := M 2 × · · · × M n . By the inductive hypothesis, C β (L, G) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. By the induction base, C k (M 1 , C β (L, G)) admits a canonical smooth manifold structure making it a C r -regular Lie group. Since C β (L, G) is canonical, the group homomorphism is a bijection (see (4.13)). We endow with the smooth manifold structure turning into a C ∞ -diffeomorphism. By Lemma 6.4, this structure is pre-canonical, makes C α (M, G) Lie group, and is compatible with evaluations.
The Lie group C α (M, G) is C r -regular, as is an isomorphism of Lie groups. Let C 1 , . . . , C be the connected components of M 1 . Let K be the set of compact, full submanifolds K of M 1 . Then, the interiors K o cover M 1 (as the interiors of connected, compact full submanifolds cover each connected component of M 1 , by the proof of Lemma 6.8). Now C k (K , G) admits a canonical smooth manifold structure making it a C r -regular Lie group, by Lemma 6.7. Thus, C β (L, C k (K , G)) admits a canonical smooth manifold structure, by the inductive hypothesis. By Lemma 6.9, the pre-canonical smooth manifold structure on C α (M, G) is canonical.
Step 2 (the general case). Let M 1 , . . . , M n be arbitrary. Using Lemma 4.12 (a), we may reorder the factors and assume that there exists an m ∈ {0, . . . , n} such that M j is compact for all j ∈ {1, . . . , n} with j ≤ m, while M j is 1-dimensional with finitely many components for all j ∈ {1, . . . , n} such that j > m. If m = 0, we have the special case just settled. If m = n, then all conclusions hold by Lemma 6.7. We may therefore assume that 1 ≤ m < n. We abbreviate K := M 1 × · · · × M m and N := M m+1 × · · · × M n . Let γ := (α 1 , . . . , α m ) and β := (α m+1 , . . . , α n ). By Step 1, C β (N , G) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. By Lemma 6.7, C β (N , G)) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. We give C α (M, G) = C γ,β (K × N , G) the smooth manifold structure making the bijection By Lemma 6.4, this smooth manifold structure is pre-canonical, makes C α (M, G) a Lie group, and is compatible with evaluations. The Lie group C α (M, G) is C r -regular as is an isomorphism of Lie groups. Now C γ (K , G) admits a canonical smooth manifold structure, which makes it a C r -regular Lie group (Lemma 6.7). By Step 1, C β (N , C γ (K , G)) admits a canonical smooth manifold structure. The pre-canonical smooth manifold structure on C α (M, G) is therefore canonical, by Lemma 6.9.
The following result complements Theorem 1.3. Under a restrictive hypothesis, it provides a Lie group structure without recourse to regularity.
is an isomorphism of topological vector spaces. For x ∈ M, let ε x : H → G and e x : C α (M, g) → g be the respective point evaluation at x. We show that β(v) = (L(ε x )(v)) x∈M for each v ∈ h, whence the Lie group structure on H is compatible with evaluations. Regard v = [γ ] as a geometric tangent vector. As L(ε x )(v) ∈ g, we have and e x is continuous and linear. For the final assertion, see Lemma 6.5.

Manifolds of maps with finer topologies
We now turn to manifold structures on C α (M, N ) for non-compact M, which are modeled on suitable spaces of compactly supported C α -functions. Notably, a proof for Theorem 1.4 will be provided. Such manifold structures need not be compatible with the compact-open C α -topology, and need not be pre-canonical. But we can essentially reduce their structure to the case of canonical structures for compact domains, using box products of manifolds as a tool. We recall pertinent concepts from [14].

7.1
If I is a non-empty set and (M i ) i∈I a family of C ∞ -manifolds modeled on locally convex spaces, then the fine box topology O fb on the Cartesian product P := i∈I M i is defined as the final topology with respect to the mappings φ : for φ := (φ i ) i∈I ranging through the families of charts φ i : = i∈I E i is endowed with the locally convex direct sum topology, and the left-hand side V φ of (9), which is an open subset of E φ , is endowed with the topology induced by E φ . Let U φ := φ (V φ ). Thus, Note that the projection pr i : P → M i is continuous for each i ∈ I , entailing that the fine box topology is Hausdorff. In fact, using the continuous linear projection π i : E φ → E i onto the ith component, we deduce from the continuity of pr i • φ = φ −1 i • π i | V φ for each φ that pr i is continuous.

7.2
Let φ be as before and ψ be an analogous family of charts ψ i : for all but finitely many i ∈ I , then which is an open subset of i∈I E i . The transition map is C ∞ (as follows from [11,Proposition 7.1]) and in fact a C ∞ -diffeomorphism, and hence a homeomorphism, since −1 ψ • φ is the inverse map. If φ −1 i (0) = ψ −1 i (0) for infinitely many i ∈ I , then ( φ ) −1 (U φ ∩ U ψ ) = ∅ and the transition map trivially is a homeomorphism. Using a standard argument, we now deduce that U φ = φ (V φ ) is open in (P, O fb ) for all φ and φ is a homeomorphism onto its image (see, e.g., [15, Exercise A.3.1]). By the preceding, the maps φ := ( φ | U φ ) −1 : U φ → V φ ⊆ E φ are smoothly compatible and hence form an atlas for a C ∞ -manifold structure on P. Following [14], we write P fb for P, endowed with the topology O fb and the smooth manifold structure just described, and call P fb the fine box product.
Some auxiliary results are needed. We use notation as in 5.8 and Theorem 1.4.
(a) If M 1 , . . . , M n are compact, then the following bilinear map is continuous:  N) and it can be omitted without affecting the initial topology. The topology on f ,K is therefore initial with respect to ρ, and hence also with respect to the co-restriction r of ρ. Thus, r is a topological embedding and hence a homeomorphism, as r (τ ) = σ can be achieved Being linear, r is an isomorphism of topological vector spaces.
Proof of Theorem 1.4. For j ∈ {1, . . . , n}, let (K j,i ) i∈I j be a locally finite family of compact, full submanifolds K j,i of M j whose interiors cover M j . Let I := I 1 × · · · × I n . Then, the sets K i := K 1,i 1 ×· · ·× K n,i n form a locally finite family of compact full submanifolds of M whose interiors cover M, for i = (i 1 , . . . , i n ) ∈ I . The map (10) In fact, the inclusion "⊆" is obvious. If ( f i ) i∈I is in the set on the right-hand side, then a piecewise definition, f (x) For each i ∈ I , endow C α (K i , N ) with the canonical smooth manifold structure, as in Theorem 1.1, modeled on the set { f : f ∈ C α (K i , N )} of the locally convex spaces f : N ). Let : T N ⊇ U → N be a local addition for N ; as in Sect. 5, write U := {(π T N (v), (v)) : v ∈ U } and θ := (M, N ), let f be the set of all τ ∈ C α (M, T N) such that π T N • τ = f and U ) and g| M\K = f | M\K for some compact subset K ⊆ M. ( f , g). The linear map is continuous on f ,L for each compact subset L ⊆ M (see Lemma 7.3 (b)) and hence continuous on the locally convex direct limit f . As above, we see that which is a closed vector subspace of i∈I f | K i . We now show that s is a homeomorphism onto its image. In fact, s admits a continuous linear left inverse. To see this, pick a C ∞ -partition of unity (h i ) i∈I on M subordinate to (K o i ) i∈I ; then L i := supp(h i ) is a closed subset of K i and thus compact. The multiplication operator 7.3 (a)). Moreover, the restriction operator s i : f ,L i → f | K i ,L i is an isomorphism of topological vector spaces (Lemma 7.3 (c)). Thus, s −1 i •β i : f | K i → f ,L i ⊆ f is a continuous linear map. By the universal property of the locally convex direct sum, also the linear map σ : is continuous. Hence, σ | im(s) is continuous and linear. We easily verify that σ • s = id f .
In fact, for i, j ∈ I and x ∈ K i ∩ K j we have (τ i (x)) = (τ j (x)) if and only if τ i (x) = τ j (x), from which the assertion follows in view of (10) and (11). Thus, showing that im(ρ) is a submanifold of fb i∈I C α (K i , N ). Let be the corresponding submanifold chart for im(ρ). Then, Note that the smooth manifold structure on C α (M, N ) which is modeled on E and makes ρ a C ∞ -diffeomorphism is uniquely determined by these properties. Thus, it is independent of the choice of . On the other hand, the (φ f ) −1 form a C ∞ -atlas for a given local addition . As the definition of the φ f does not involve the cover (K i ) i∈I , the smooth manifold structure just constructed is independent of the choice of (K i ) i∈I .
The following lemma fills in the details for 3.3.
is a C α−ke n -map for all k ∈ N 0 such that k ≤ α n .
Proof We show by induction on k 0 ∈ N that the conclusion holds with k ≤ k 0 for all functions as described in the lemma, for all α with α n ≥ k 0 . Using local charts, we may assume that U j := M j is a locally convex subset of a locally convex space E j for all j ∈ {1, . . . , n} and N a locally convex subset of a locally convex space F; thus f is a map U := U 1 ×· · ·×U n → F. The case k 0 = 0 being trivial as h 0 = f is C α . Let 1 ≤ k 0 ≤ α n now. Then, d e n f : U 1 × · · · × U n × E n → F is a C (α−e n ,0) -map. Being linear in the final argument, d e n f is C α−e n as a map of n variables, i.e., as a map on the domain T e n U = U 1 ×U n−1 ×T U n (see [1,Lemma 3.11]). Let pr 1 : T U n = U n × E n → U n be the projection onto the first component. Then, g := f • id U 1 × · · · × id U n−1 × pr 1 : U 1 × · · · × U n−1 × T U n → F is C α by the chain rule [1,Lemma 3.16], and hence C α−e n . Thus, h 1 = (g, d e n f ) is C α−e n , by [1,Lemma 3.8]. By the inductive hypothesis, the maps are C α−e n − je n for all j ∈ {0, . . . , k 0 − 1}. It only remains to observe that this map equals h j+1 .

Proof of Lemma 3.5. (a)
For β ∈ N n 0 with β ≤ α, consider the maps Going through the recursive construction of T β (g • f ) in 3.3 for f ∈ C α (M, N ) and making repeated use of the functoriality of T , we see that Thus, τ β • C α (M, g) = C(T β M, T |β| g) • T β , which is a continuous map by [15, Lemma A.6.3]. The topology on C β (M, L) being initial with respect to the maps τ β , we deduce that C α (M, g) is continuous. (b) For β ∈ N n 0 with β ≤ α, consider the maps T β : C α (M, N ) → C(T β M, T |β| N ), f → T β f and τ β : C α (L, N ) → C(T β L, T |β| N ), f → T β f . Going through the recursive construction of T β ( f • g) in 3.3 for f ∈ C α (M, N ) and making repeated use of the functoriality of T , we see that with h β := T β 1 g 1 × · · · × T β n g n . Thus, τ β • C α (g, N ) = C(h β , T |β| N ) • T β , which is a continuous map by [15, Lemma A.6.9]. The topology on C α (L, N ) being initial with respect to the maps τ β , we deduce that C α (g, N ) is continuous. The compact-open C α -topology on C α (K i , N ) being initial with respect to the mappings τ β,i : C α (K i , N ) → C(T β K i , T |β| N ), f → T β f , we deduce from ρ β,i • τ β = τ β,i • ρ that O is initial with respect to the maps ρ i .
Proof of Lemma 3.8. The case n = 1 is well known. The general case follows as T β S = T β 1 S 1 × · · · × T β n S n and T β M = T β 1 M 1 × · · · × T β n M n .
Proof of Lemma 3.9. The inclusion map λ : S → N is smooth. By Lemma 3.8, the inclusion map T |β| λ : T |β| S → T |β| N is a topological embedding, for each β ∈ (N 0 ) n such that β ≤ α. is a linear map. It is a homeomorphism onto its image, which is a locally convex space. Hence also, C α (M, F) is a locally convex space.
Proof of Lemma 3.11. (a) For each k ∈ N 0 , the topology on T k F = F 2 k is initial with respect to the linear maps T k λ i = λ 2 k i : F 2 k → F 2 k i . For each β ∈ N n 0 with β ≤ α, the compact-open topology on C(T β M, T |β| F) is therefore initial with respect to the mappings The topology on C α (M, F i ) being initial with respect to the mappings τ i,β : C α (M, F i ) → C(T β M, T |β| F i ) for β ≤ α, we deduce that O is initial with respect to the mappings C α (M, λ i ) = (λ i ) * .
(b) By [1,Lemma 3.8], the linear map is a bijection. The topology on F being initial with respect to the maps pr i , (a) shows that the topology on C α (M, F) is initial with respect to the maps (pr i ) * and hence makes a topological embedding. Hence, is a homeomorphism, being bijective. (c) By [1,Lemma 3.8], is a bijection. By Lemma 3.5, is continuous. To see that −1 is continuous, we prove its continuity at a given element ( f 1 , f 2 ) in C α (M, N 1 ) × C α (M, N 2 ). For x ∈ M, pick a chart φ x,i : U x,i → V x,i ⊆ E x,i of N i around f i (x), for i ∈ {1, 2}. There exist compact full submanifolds K x, j of M j for j ∈ {1, . . . , n} such that K x := K x,1 × · · · × K x,n ⊆ ( f 1 , f 2 ) −1 (U x,1 × U x,2 ) and x ∈ K o x . By Lemma 3.7, the topology on C α (M, N 1 × N 2 ) is initial with respect to the restriction maps It thus suffices to show that ρ x • −1 is continuous at ( f 1 , f 2 ) for all x ∈ M. Now ρ x • −1 = −1 x • (ρ x,1 × ρ x,2 ) using the continuous restriction maps ρ x,i : C α (M, N i ) → C α (K x , N i ) for i ∈ {1, 2} and the map taking a function to its pair of components. Thus, it suffices to show that −1 x is con- N i ), on which the latter induces the compact-open C α -topology, by Lemma 3.9. The map −1 takes this set onto C α (M, U x,1 × U x,2 ), on which C α (M, N 1 × N 2 ) induces the compact-open C α -topology. It thus suffices to show that −1 x is continuous at ( f 1 | K x , f 2 | K x ) as a map C α (K x , U x,1 ) × C α (K x , U x,2 ) → C α (K x , U x,1 × U x,2 ). Now (φ x, j ) * : C α (K x , U x, j ) → C α (K x , V x,i ) is a homeomorphism for i ∈ {1, 2} and also (φ x,1 ×φ x,2 ) * : C α (K x , U x,1 ×U x,2 ) → C α (K x , V x,1 × V x,2 ) is a homeomorphism, by Lemma 3.5. It thus suffices to show that the mapping (φ x,1 ×φ x,2 ) * • −1 x •((φ x,1 ) * × (φ x,2 ) * ) −1 : is continuous. But this mapping is a restriction of the homeomorphism C α (K x , E x,1 ) × C α (K x , E x,2 ) → C α (K x , E x,1 × E x,2 ) discussed in (b).
is a bijective linear map; by Lemma 3.7, it is a homeomorphism. Likewise, and R : C α (N , C β (M, E)) → i∈I C α (U i , C β (M, E)), f → ( f | U i ) i∈I are isomorphisms of topological vector spaces. By Lemma 3.5, the mapping C α (U i , r ) : C α (U i , C β (M, E)) → C α (U i , j∈J C β (V j , E)) is an isomorphism of topological vector spaces and so is the map taking a map to its family of components (see Lemma 3.11 (b)). Hence, is an isomorphism of topological vector spaces. By [1,Theorem B], the map i, j : E)), f → f ∨ is linear and a topological embedding, whence so is Evaluating at x ∈ N and then in y ∈ M (say x ∈ U i and y ∈ V j ), we verify that (N , C β (M, E)) and makes sense as a map to the latter space. We have a commutative diagram where the vertical arrows are homeomorphisms and is a topological embedding. Hence, is a topological embedding. If M is locally compact, then so are the V j , whence each of the maps i, j is a homeomorphism by [1,Theorem 4.4] and hence also . Then also, = −1 • • ρ is a homeomorphism.