Appendix A: Bastiani Calculus

The notion of calculus in locally convex spaces we need is the so-called Bastiani calculus, and its origin can be traced back to [2, 37]. Here, we introduce the basic definitions and properties.

In the sequel, we will use complete locally convex spaces and denote them by capital letters X, Y, Z.

Definition A.1

Let \(U\subset X\) be an open subset, a mapping \(P:U\subset X \rightarrow Y\) is Bastiani differentiable if the following conditions hold:

(i):

\( \lim _ \frac\big (P(x+tv)-P(x)\big )=dP[x](v)\) exists for all \(x\in U\), \(v\in X\), giving rise to a mapping \(dP:U\times X \rightarrow Y\) linear in the second entry;

(ii):

The mapping \(dP:U\times X \rightarrow Y, \ (x,v) \mapsto dP[x](v)\) is jointly continuous.

Theorem A.2

Let \(\gamma :[a,b]\subset } \rightarrow X\) be a continuous curve, then there exists a unique object \(\int _a^b \gamma (t)\text t\in X\) such that

(i):

for every continuous linear mapping \(l:X\rightarrow }\)

$$\begin l\bigg (\int _a^b\gamma (t)\text t\bigg ) =\int _a^bl(\gamma (t))\text t; \end$$

(ii):

for every seminorm \(p_i\) on X,

$$\begin p_i\bigg (\int _a^b\gamma (t)\text t\bigg ) \le \int _a^bp_i(\gamma (t))\text t; \end$$

(iii):

for all continuous curves \(\gamma \), \(\beta \), \(\int _a^b \big (\gamma (t)+ \beta (t)\big )\text t=\int _a^b \gamma (t)\text t+\int _a^b \beta (t)\big )\text t\);

(iv):

for all \(\lambda \in }\), \(\int _a^b \big (\lambda \gamma (t)\big )\text t=\lambda \bigg (\int _a^b \gamma (t)\text t\bigg )\);

(v):

for all \(a\le c \le b\), \(\int _a^b\gamma (t) \text t=\int _a^c\gamma (t) \text t +\int _c^b\gamma (t) \text t\).

The proof of this result is quite standard (see, e.g., [30, Theorem 2.2.1, pp. 71]).

Lemma A.3

Let \(U\subset X\), \(V\subset Y\) be open subsets, and let \(P:U\rightarrow Y\) and \(Q:V\rightarrow Z\) be Bastiani differentiable mappings such that \(P(U) \subset V\), then \(Q\circ P : U \rightarrow Z\) is Bastiani differentiable and \(d(Q\circ P)[x](v)= dQ[P(x)]\big (dP[x](v)\big )\) for each \(x\in U\), \(v\in X\).

Proof

We claim that P is Bastiani differentiable if and only if there is a continuous mapping \(L:U\times U \times X \rightarrow Y\) linear in the third entry such that

$$\begin P(x_1)-P(x_2)= L[x_1,x_2](x_1-x_2). \end$$

In particular, we have \(L[x,x](v)=dP[x](v)\). The necessity condition follows by considering the smooth curve \(\gamma (t)=x_1+t(x_2-x_1)\) with \(t\in [0,1]\), then \(dP[\gamma (t)](v)\) is a smooth curve for each \(v\in X\), and by A.2, define

$$\begin L[x_1,x_2](v)\doteq \int _0^1 dP[\gamma (t)](v)\text t. \end$$

It is clear that L is continuous and linear in the third entry, and moreover, since \(\frac}t}P(\gamma (t))=dP[\gamma (t)](x_1-x_2)\), we get \(L[x_1,x_2](x_1-x_2)= dP[x_1](x_1-x_2)\). For the sufficiency condition, just note that \(\frac\big (P(x+tv)-P(x)\big )=L(x, x+tv)[v]\), so taking the limit we get our claim. Next suppose that

$$\begin P(x_1)-P(x_2)= & L[x_1,x_2](x_1-x_2),\\ Q(y_1)-Q(y_2)= & M[y_1,y_2](y_1-y_2). \end$$

Then, \(Q(P(x+tv))-Q(P(x))=M[P(x+tv),P(x)]\big (P(x+tv)-P(x)\big )=t M[P(x+tv),P(x)]\big (L[x+tv,x](v)\big )\), thus dividing by t and taking the limit yields \(d(Q\circ P)[x](v) \equiv M[P(x),P(x)]\big (L[x,x](v)\big )= dQ[P(x)]\big (dP[x](v)\big )\). \(\square \)

Definition A.4

A mapping \(f:U\subset X \rightarrow Y\) is k times Bastiani differentiable if \(d^f:U\times X \cdots \times X \rightarrow Y\) is Bastiani differentiable. The kth derivative of f at x is defined by recursion

$$\begin d^f[x](v_1,\ldots ,v_k)\doteq & \lim _ \frac\big ( d^f[x+tv_k](v_1,\ldots ,v_)\nonumber \\ & -d^f[x+tv_k](v_1,\ldots ,v_)\big ) . \end$$

(79)

Explicit computation of \(\text \big ( d^f\big )\) in \((x,v_1,\ldots ,v_)\) yields

$$\begin \begin&\text \big (d^f\big )[x,v_1,\ldots ,v_](v_k,w_1,\ldots ,w_)\\&\quad = \sum _^ d^f[x](v_1,\ldots , \widehat, w_j, \ldots , v_)\\ &\qquad + \ \lim _ \frac\big ( d^f[x+tv_k](v_1,\ldots ,v_)-d^f[x+tv_k](v_1,\ldots ,v_)\big ) , \end \end$$

and we can then see that \(\text \big ( d^f\big )\) is Bastiani differentiable if and only if the limit of (79) exists and is a continuous mapping \(U\times X^k \rightarrow Y\). We can thus state

Lemma A.5

A mapping \(f : U \subset X \rightarrow Y\) is k times Bastiani differentiable if and only if for each \(0\le j \le k\) all the derivative mappings \(d^f:U \times X^j \rightarrow Y\) exist and are jointly continuous.

Appendix B: Topologies on Spaces of Mappings

Let M, N be finite-dimensional paracompact Hausdorff topological spaces, denote the space of continuous functions by C(M, N). The compact-open topology \(\tau _\) or CO-topology is the topology generated by a basis whose elements have the form

$$\begin N(K,V)=\, \end$$

(80)

where \(K\subset M\) is a compact subset and \(V\subset N \) is open. Roughly speaking, this topology controls the behavior of functions only on small regions of M, whereas their behavior “at infinity” is not specified.

Lemma B.1

Let M, N as described above, if N is normal, then \(\big ( C(M,N), \tau _\big )\) is Hausdorff.

Proof

Supposing \(\varphi \ne \psi \), then at least \(\varphi (x)\ne \psi (x) \) for some \(x\in M\). By continuity of \(\varphi \), \(\psi \) there exists an open subset \(U_x\) such that \(\varphi (y)\ne \psi (y)\) for each \(y\in }_x\). Without loss of generality, we can suppose that \(\overline\) is compact, and then, \(\varphi (\overline)\), \(\psi (\overline)\) are compact and therefore closed. Since N is normal, there are disjoint open subsets \(V_\), \(V_\), respectively, containing \(\varphi (\overline)\), \(\psi (\overline)\), then

$$\begin N\big (U_x,V_ \big )\cap N\big (U_x,V_ \big )=\emptyset . \end$$

\(\square \)

Lemma B.2

Let M, N be topological spaces, if N is a complete metric space, then \(\big ( C(M,N), \tau _\big )\) is a complete metric space as well.

If M is compact and \((N,d_N)\) is metric, then a neighborhood of \(\varphi \) in the compact-open topology can be given as

$$\begin B_(\varphi )\doteq \, \end$$

where \(\epsilon :M \rightarrow }_+\) is a continuous function.

Given \(\varphi \in C(M,N)\), let \(G_:M\rightarrow M\times N\) be the graph mapping associated with \(\varphi \), set \(\textrm(G_)\equiv \textrm(\varphi )=\\).

Definition B.3

The wholly open topology \(\tau _}\) or \(\textrm\)-topology on C(M, N) is generated by a subbasis of open subsets of the form

$$\begin W(V)=\, \end$$

(81)

where \(V\subseteq N\) is open.

Note that the \(\textrm\)-topology is not Hausdorff, for it cannot separate surjective functions.

Definition B.4

The graph topology \(\tau _^0}\) or \(}^0\)-topology on \(C^(M,N)\) is the one induced by requiring

$$\begin G:C(M,N) \ni \varphi \mapsto G_ \in \big ( C(M,M\times N),\tau _}\big ) \end$$

to be an embedding.

By Definition B.3, the open subbasis of \(C(M,M\times N)\) is given by subsets of the form

$$\begin W(}) = \}\} \end$$

with \(}\subset M\times N\) open subsets. When \(f=G_\) for some \(\varphi \in C(M,N)\), then the trace topology on the subset \(G\big (C(M,N)\big )\) is generated by a subbasis of elements \(W(})\) where \(}=M\times V\) with \(V\subset N\) open subset. Clearly, G is an injective mapping, and bijective onto its image. Therefore, a subbasis for the \(\textrm^0\)-topology is given by

$$\begin W(})= \\subset M\times V\} \end$$

(82)

Lemma B.5

The \(\textrm^0\)-topology is finer than the CO-topology and is therefore Hausdorff.

Proof

We show that \(\textrm_: \big ( C(M,N),\tau _^0}\big ) \rightarrow \big ( C(M,N),\tau _}\big )\) is continuous. Let N(K, V) be an open subset as in (80), and \(U_1\), \(U_2\) be a cover of M such that \(K\subset U_1\) and \(U_2 =M\backslash K\). Consider the open subset

$$\begin W(U_1\times V \cup U_2 \times N)=\\subset U_1\times V \cup U_2 \times N\}; \end$$

the former is a \(\textrm^0\)-open subset, which is, however, equal to N(K, V). \(\square \)

The main difference from the compact-open topology is that the \(\textrm^0\) topology does control the behavior of a mapping over the whole space, while the \(\textrm\) topology was limited to a compact region. Notice that although in general the \(\textrm^0\) topology is finer than the \(\textrm\) topology, when the manifold M is compact, the two become equivalent.

Lemma B.6

Let M be paracompact and (N, d) be a metric space, then a basis of neighborhood of \(\varphi \in C(M,N)\) for the \(\textrm^0\)-topology is given by

$$\begin W_(\epsilon )=\big \\big (\varphi (x),\psi (x)\big )<\epsilon (x) \ \forall x\in M\big \}, \end$$

(83)

where \(\epsilon :M \rightarrow }_+\) is continuous.

Proposition B.7

Let M be paracompact and (N, d) be a metric space, then for any sequence \(\\subset C(M,N)\), the following are equivalent:

(i):

\(\varphi _n \rightarrow \varphi \) in the \(\textrm^0\)-topology;

(ii):

there exists a compact set \(K\in M\) such that \(\varphi _n\big \vert _\equiv \varphi \big |_\) for each \(n\in }\), and \(\varphi _n \rightarrow \varphi \) uniformly on K.

Notice that, due to Proposition B.7, the space C(M, E) with E vector space is not a topological vector space, in particular the multiplication mapping cannot be continuous since if \(\lambda \in } \) goes to 0, then \(\lambda \cdot f \not \rightarrow 0\) unless \(f=0\) outside some compact subset of M.

Proof

Suppose \(\varphi _n \rightarrow \varphi \) in the \(\textrm^0\)-topology, however, for all \(K\subset M\) compact, either \(\varphi _n \not \rightarrow \varphi \) uniformly over K or there is \(x\in M\backslash K\) such that \(\varphi _n(x) \ne \varphi (x)\). In the first case \(\varphi _n \not \rightarrow \varphi \) in the CO-topology as well, contradicting the initial hypothesis. In the second case, let \(\\) be an exhaustion of compact subsets of M, then for each \(n\in }\) there is \(x_n \in M\backslash K_n\) having \(\varphi _n(x_n)\ne \varphi (x_n)\). Set \(0<\epsilon _n =\sup _d(\varphi _n(x),\varphi (x))\). For each n, consider the sequence of open neighborhoods of \(\varphi \), \(W_(\epsilon _n)\) as per Lemma B.6, by construction \(\varphi _n \notin W_(\epsilon _n)\) which contradicts the convergence hypothesis. On the other hand, let \(\epsilon _n:M \rightarrow }_+\) be the constant functions with \(\epsilon _n= \sup _d(\varphi _n(x),\varphi (x))\), then \(W_(\epsilon _)\cap \=\_\) by uniform convergence over K. Implying \(\varphi _n\rightarrow \varphi \) in \(\tau _^0}\). \(\square \)

Corollary B.8

Let M and N as in Proposition B.7 and \(\gamma :I\subset }\rightarrow \big (C(M,N),\tau _^0}\big )\) be a continuous mapping with I compact. Then, there exists a compact \(K\subset M\) such that

$$\begin \gamma (t) : x \in M \rightarrow N \end$$

is constant in \(M\backslash K\) for each \(t\in I\).

Proof

We argue by contradiction, let \(K_n\) be an exhaustion of compact subsets of M, then for each \(n\in }\) there is some \(t_n\in I\), and some \(x_n\in M\backslash K_n\) such that \(\gamma (t_n)[x_n]\ne \gamma (t)[x_n]\) for at least a \(t\in I\). Since \(\\) is a sequence on a compact space, we may assume, eventually passing to a subsequence, that \(t_n\rightarrow t_0\in I\), by construction, \(\\) does not admit a cluster point in M. Finally, by continuity, \(t_n \rightarrow t_0 \implies \gamma (t_n)\rightarrow \gamma (t)\) in the \(\textrm^0\)-topology, by Proposition B.7 there has to be a compact subset K such that \(\gamma (t_n)\equiv \gamma (t)\) outside K thus the sequence \(\\) admits a cluster point. \(\square \)

From now on, we assume that M, N are smooth m, n-dimensional manifolds, respectively, consider the kth-order jet bundle \(J^k(M,N)\). Recall as well the mappings \(\alpha :J^k(M,N) \rightarrow M\), \(\beta :J^k(M,N) \rightarrow N\).

Definition B.9

The Whitney \(\textrm^k\)-topology, or \(\textrm^k\)-topology, on \(C^r(M,N)\) for \(0\le k\le r \le \infty \) is the topology induced by requiring

$$\begin j^k:C^r(M,N) \rightarrow \Big ( C\big (M,J^k(M,N)\big ),\tau _^0}\Big ) \end$$

to be topological embedding.

Proposition B.10

The \(\textrm^k\)-topology on \(C^r(M,N)\) enjoys the following properties:

(i):

A subbasis of open subsets of the topology has the form

$$\begin W(})=\}\} \end$$

(84)

where \(}\subset J^k(M,N)\) is an open subset.

(ii):

If \(d_k\) is a metric on \(J^k(M,N)\), then a basis of neighborhoods for the \(\textrm^k\)-topology of \(\varphi \in C^r(M,N)\) is

$$\begin N_^k(\epsilon )=\ \end$$

where \(\epsilon \in C(M,}_+)\).

(iii):

The sequence \(\\subset C^k(M,N)\) converges to \(\varphi \) in the \(\textrm^k\)-topology if and only if there is a compact subset \(K\subset M\) such that \(\varphi _n\equiv \varphi \) in \(M\backslash K\) and \(j^k\varphi _n \rightarrow j^k\varphi \) uniformly over K.

(iv):

If \(I\subset }\) is compact and \(\gamma :I \rightarrow \big ( C^r(M,N), \tau _^k} \big )\) is continuous, then there is a compact subset such that

$$\begin \textrm_x\gamma : I\ni t \mapsto \gamma (t)[x] \end$$

is constant for all \(x\in M\backslash K\).

(v):

\(\textrm^\) on \(C^(M,N) \) is the projective limit topology of all \(\textrm^k\)-topologies for \(0\le k\le \infty \).

(vi):

A basis of open neighborhood of the \(\textrm^\) topology on \(C^(M,N)\) consists of open subsets

$$\begin W(})=\(M,N): j^\varphi (M)\subset }\} \end$$

(85)

where \(}\subset J^(M,N)\) is open.

(vii):

If \(\_n\) is an exhaustion of compact subsets of M, a basis for the \(\textrm^\) topology on \(C^(M,N)\) consists of open subsets

$$\begin M(U,n)= \(M,N): j^nf(M\backslash K_n^})\subset U_n\} \end$$

where \(U_n \subset J^n(M,N)\) are open.

Proof

We claim that for the mapping \(j^k : C^(M,N) \rightarrow C(M, J^k(M,N))\) the \(\textrm^0\) and \(\textrm\) topologies on \(j^k\big (C^(M,N)\big )\subset C(M,J^k(M,N))\) coincide. Indeed,

$$\begin G_ (M)\subset M\times J^(M,N)=\\simeq J^k(M,N), \end$$

where the last is a topological embedding, therefore open subsets of \(J^k(M,N)\) and \(G_\subset M \times J^k(M,N)\) coincide. As a result, we obtain (84) by combining the above result with (81). (ii) follows by combining (i) with Lemma B.6. Using that \(\varphi _n\rightarrow \varphi \) in \(\textrm^k\) if and only if \(j^k\varphi _n\rightarrow j^k\varphi \) in \(\textrm^0\) over \(C(M,J^k(M,N))\) in conjunction with Proposition B.7 we get (iii). Similarly, (iv) is obtained by combining Corollary B.8 with the above argument. The argument for (v) and (vi) is the following: The topology on \(J^(M,N)\) is the coarsest such that each \(\pi ^_k: J^(M,N) \rightarrow J^k(M,N)\) is continuous. Note that we have an embedding

$$\begin J^(M,N)\simeq M\times _MJ^(M,N) \hookrightarrow M\times J^(M,N). \end$$

Therefore, we construct the following commutative diagram

where the horizontal mapping are embeddings. Then, a subbasis of \(C\big ( M,J^(M,N)\big )\) for the \(\textrm^0\)-topology is

$$\begin W(U_)=\\varphi \in C^(M,N): j^(M)\subset U_\} \end$$

with \(U_\subset J^(M,N)\) open. Finally we show (vii). Let \(U_n\subset J^n(M,N)\) be open subsets, then each

$$\begin M(U)= \(M,N): \ \forall n\in },\ j^\varphi (M\backslash K_n^})\subset (\pi ^_n)^U_n\}; \end$$

is an open subset of the \(\textrm^\) topology. Setting \(V_n=(\pi ^_0)^(U_0)\cap \cdots \cap (\pi ^_n)^(U_n)\) we have that

$$\begin \(M,N): \ \forall n\in },\ j^\varphi (K_\backslash K_n^})\subset V_n\}= M(U). \end$$

The inclusion \(\supset \) is clear; for the other, observe that in each region \(K_\backslash K_n^}\) we have the requirement \( j^\varphi (K_\backslash K_n^})\subset V_n\) for all n which is way stronger then the corresponding \( j^\varphi (M\backslash K_n^})\subset (\pi ^_n)^U_n\) for all n. Since \(J^(M,N)\) is a fiber bundle with finite-dimensional base and Fréchet space fiber and has the coarsest topology making each \(\pi ^_k\) continuous, we may write

$$\begin M(})=\(M,N): \ \forall n\in },\ j^\varphi (K_\backslash K_n^})\subset }_n\}. \end$$

where each \(}_n\subset J^(M,N)\) is open. We claim that \(M(})\) generates a topology equivalent to the \(\textrm^\) topology. The latter’s open subsets possess the form (85) is thus clear that \(M(})\subset W(\cup _n}_n)\), thus making the former topology finer then the latter. To see the converse observe that \(j^\varphi (K_\backslash K_n^})\) is a compact subset of a metric space for each n, thus there is some \(\epsilon _n>0 \) for which the open subset \(\_x\psi \in }^ :d(j^_x\varphi ,j^_x\psi )<\epsilon _n \ \forall x\in K_\backslash K_n^} \} \subset }_n\). Let then \(\epsilon \in C(M)\) be a continuous function such that \(\epsilon (x)<\epsilon _n\) for all \(x\in K_\backslash K_n^}\), then \(N^_(\epsilon )\) is an open subset of the Whitney topology which is contained in \(M(})\). \(\square \)

As a consequence of Proposition B.7, when N is metrizable and M paracompact and second countable, given any sequence \(\lbrace \varphi _\rbrace _}}\) we can characterize its convergence in the following way:

(i):

\(\varphi _n \rightarrow \varphi \) in the \(\textrm^-\)topology ,

(ii):

\(\forall n'\in } \ \exists K_ \subset M\) compact such that if \(n\ge n'\) then \(\left. \varphi _n \right| _}=\left. \varphi \right| _}\) and \(\left. \varphi _n \right| _ \rightarrow \left. \varphi \right| _\) uniformly with all its derivatives.

This fact has important implications; for if we consider the finite-dimensional vector bundle \(E \rightarrow M\), the vector space \(\Gamma ^(M\leftarrow E)\) will not be a topological vector space due to the failure of continuity for the multiplication by scalar. This can be readily seen from condition (ii) above: If, for instance, we had \(\sigma \in \Gamma ^(M\leftarrow E)\), \(} \ni \epsilon _n \rightarrow 0\) and \(\epsilon _n\sigma \rightarrow 0\), then each \(\epsilon _n\sigma \) must possess compact support, and thus, \(\sigma \) itself ought to be compactly supported. As a consequence, we get the following result:

Theorem B.11

Let \((E,\pi ,M)\) be a finite-dimensional vector bundle, then \(\Gamma ^_c(M\leftarrow E)\subset \Gamma ^(M\leftarrow E)\), equipped with trace of the Whitney topology on \(C^(M,E)\). Then, \(\Gamma ^_c(M\leftarrow E)\) is the maximal locally convex space contained in \(\Gamma ^(M\leftarrow E)\). Moreover, the trace topology coincides with the (natural) final topology induced by the projective limit

$$\begin \lim _ \longrightarrow \\ K\subset M \end}\Gamma ^_K(M\leftarrow E) =\Gamma ^_c(M\leftarrow E). \end$$

(86)

Consequently, \(\Gamma ^_c(M\leftarrow E) \) is a complete, nuclear and Lindelöf space, hence paracompact and normal. In particular, for each open cover \(}_i\) of \(\Gamma ^_c(M\leftarrow E)\), there are Bastiani smooth bump functions \(\rho _i:\Gamma ^_c(M\leftarrow E)\rightarrow }\) each of which has \(\textrm(\rho _i)\subset }_i\), satisfying

$$\begin \sum _i\rho _i(\sigma )=1 \end$$

for each \(\sigma \in \Gamma ^_c(M\leftarrow E)\).

From a topological standpoint, Theorem B.11 implies that \(\Gamma ^_c(M\leftarrow E)\subset \Gamma ^(M\leftarrow E)\) equipped with the Whitney topology is the maximal topological vector subspace. Thus, if we want to give a topological manifold structure to spaces such as \(\Gamma ^(M\leftarrow E)\) (or \(C^(M,N)\)), this forces us to use \(\Gamma ^_c(M\leftarrow E)\) as the topological vector space on which to model the manifold (see Definition C.1). It is therefore natural to seek charts of the form \((\sigma _0+\Gamma ^_c(M\leftarrow E),u_)\) where \(u_(\sigma )=\sigma -\sigma _0\). The undesirable fact is that \(\sigma _0+\Gamma ^_c(M\leftarrow E)\) would then become a closed subset. To remedy this problem, we refine the Whitney topology just enough to make the above subsets open. To wit consider the following equivalence class: Given M, N smooth finite-dimensional manifolds, set \(\varphi \sim \psi \) if \(\textrm_(\psi )= \overline\subset M\) is compact.

Definition B.12

The refined Whitney topology, or refined \(\textrm^\) topology, is the coarsest topology on \(C^(M,N)\) which is finer than the \(\textrm^\)-topology and for which the sets \(}_=\lbrace \psi \in C^(M,N): \psi \sim \varphi \rbrace \) are open.

The refined Whitney topology has the same converging sequences and smooth curves as the Whitney topology since the proofs of B.7 and Corollary B.8 remain essentially valid. Notice that the refinement we imposed on the topology was made by adding big open subsets, i.e., the trace topology on subspaces of the form \(\Gamma ^_c(M\leftarrow E)\) is not altered, thus the aforementioned properties remain valid. Clearly, \(\Gamma ^_c(M\leftarrow E) \subset \Gamma ^(M\leftarrow E)\) will then become open and moreover \(\Gamma ^(M\leftarrow E)\) becomes a topological affine space with model topological vector space \(\Gamma ^_c(M\leftarrow E)\). The locally convex modeling space \(\Gamma ^_c(M\leftarrow E)\) will, however, no longer be a Baire space: If \(N=}\) and \(K_n\) is an exhaustion of compact subsets of M, then \(\cup _n C^_(M)=C^_c(M)\), and, however, \(C^_(M)\subset C^_(M)\) is not dense for all \(n\in }\).

Proposition B.13

Let M, \(M'\), N, \(N'\) be smooth finite-dimensional manifolds,

(i):

if \(f:M'\rightarrow M\) is a proper smooth mapping, then \(f^*:C^(M,N) \rightarrow C^(M',N) \) is continuous in both the Whitney and refined Whitney topology;

(ii):

if \(h:N\rightarrow N'\) is a smooth mapping, then \(h_*:C^(M,N) \rightarrow C^(M,N') \) is continuous in both the Whitney and refined Whitney topology.

Proof

The proof of (i) can be directly obtained by using Proposition 7.3 in [38] while keeping in mind that \(f^*(\varphi )=\varphi \circ f\). For (ii), consider a Whitney open subset \(M'(U)=\(M,N') : \ j^n\varphi (M\backslash K_n)\}\subset U_n \subset J^(M,N') \forall n \in } \}\) in \(C^(M,N')\). The mapping \(j^nh : J^n(M,N) \rightarrow J^n(M,N')\) is smooth and hence continuous, and thus, set \(V_n=(j^nh)^(U_n)\). Then, \((h_*)^(M'(U))=M(V)\) which implies continuity of \(h_*\) in the Whitney topology. For the refined Whitney topology, one notes that if \(\varphi \sim \varphi '\), then \(h\circ \varphi \sim h \circ \varphi '\) as well, thus \(h_*}_\varphi =}_\), which in turn implies that \(h_*\) remains continuous even when refining the topology. \(\square \)

Theorem B.14

(Proposition 4.8, pp. 38 [38]). Let \((E_i,\pi _i,M)\), \(i=1,2\) be finite-dimensional vector bundles, suppose that \(\alpha : U\subset E_1\rightarrow E_2 \) is a smooth fibered morphism projecting to the identity of M and let \(\sigma _0\in \Gamma ^_c(M\leftarrow E_1)\) having \(\sigma _0(M)\subset U\), \(\alpha (\sigma _0)\in \Gamma ^_c(M\leftarrow E_2)\). Then, the mapping \(\alpha _:}=\_c(M\leftarrow E_1): \sigma (M)\subset U\} \rightarrow \Gamma ^_c(M\leftarrow E_2)\) is a Bastiani smooth mapping; moreover, if \(d_v\alpha :VU\rightarrow E_2\) is the vertical derivative of \(\alpha \), we have \(d(\alpha _*)=(d_v\alpha )_*\).

We remark that given any connection on the vector bundle \(E_1\) it induces a splitting \(TE_1=HE_1\oplus VE_1\) into horizontal and vertical vector bundle, and the latter is of course independent from the connection chosen. Thus, if we use local fibered coordinates on \(E_1\) and \(E_2\) induced by local frames \(e_i\), \(f_i\), respectively, and study \(\alpha \) in a neighborhood of \(p\in E_1\), \(\alpha (p)= \sum \alpha ^i(x,y)f_i\), then \(d_v\alpha :V_pE_1\rightarrow E_2, \ (p;\sigma ) =\sum \frac\sigma ^jf_i \in E_2|_\).

Proof

It is clear that if U is open, \(}=\_c(M\leftarrow E_1): \sigma (M)\subset U\}\) is open in \(\Gamma ^_c(M\leftarrow E_1)\) with the Whitney topology as well by (vi) of Proposition B.10 taking \(\widetilde}}=(\pi ^)^(M\times U)\). Next we show that \(\alpha _*\) is Bastiani differentiable. This is equivalent to show that \(d(\alpha _*):}\times \Gamma ^_c(M\leftarrow E_1)\rightarrow \Gamma ^_c(M\leftarrow E_2)\) exists and is continuous in the Whitney topology (see Definition B.9). We thus claim that

$$\begin \lim _ \frac = (d_v\alpha )_*(\sigma ;\sigma ') \end$$

(87)

in the Whitney topology. We start by showing that for any neighborhood U of \(x_0\in M\), \(\frac\) converges uniformly to \((d_v\alpha )_*(\sigma ;\sigma ')(x)\) in U. This is a local problem, and we can thus study it using local coordinates. Notice that if U lies outside the support of \(\sigma '\), then the claim is trivial. By an abuse of notation, we set \(\sigma (x)=(x,\sigma (x))\) and likewise for \(\sigma '\). Then by Taylor theorem, we have

$$\begin \begin&\big (\alpha _*(\sigma +t\sigma ')-\alpha _*(\sigma )\big )^i(x)\\&\quad = \alpha ^i(x,\sigma (x)+t\sigma '(x))-\alpha ^i(x,\sigma (x))\\&\quad = t\partial _j\alpha ^i(x,\sigma (x))\sigma '^j(x)\\&\qquad +t^2\int _0^1(1-\lambda )\partial _\alpha ^i(x,\sigma (x)+t\lambda \sigma '(x))\sigma '^j(x)\sigma '^k(x)\text \lambda , \end \end$$

We can then estimate in U

$$\begin \begin \bigg |\frac-\big ((d_v\alpha )_*(\sigma ;\sigma ')\big )^i(x) \end \bigg | \le |t| C_ \end$$

for each \(x\in U\), establishing uniform convergence. Moreover, since \(d_v\alpha : VU\subset VE_1 \rightarrow E_2\) remains a smooth fibered morphism, the mapping \((d_v\alpha )_*:}\times \Gamma _c^(M\leftarrow E_1)\rightarrow \Gamma _c^(M\leftarrow E_2)\) is continuous by Proposition B.10. Moreover, if \(d^k_v\alpha : \otimes ^k_MV_pU\) is the mapping locally defined by

$$\begin d^k_v\alpha [p]: \otimes ^k V_pE_1\ni (s_1,\ldots ,s_k) \mapsto \partial _\alpha ^i(p) s_1^\cdots s_k^, \end$$

a similar argument to the one above shows that for each \(x\in U\) the mapping \(d^(\alpha _*)[\sigma +t\sigma '](\sigma _1,\ldots ,\sigma _)(x)-d^(\alpha _*)[\sigma ](\sigma _1,\ldots ,\sigma _)(x)\) converges uniformly to \((d^k_v\alpha )_*(\sigma ;\sigma ',\sigma _1,\ldots ,\sigma _)(x)\). Then again, Proposition B.13 implies the continuity of \(d^(\alpha _*):}\times \Gamma _c^(M\leftarrow E_1) \cdots \times \Gamma _c^(M\leftarrow E_1)\rightarrow \Gamma _c^(M\leftarrow E_2)\). Finally, by (iii) in Proposition B.10, this shows that the mapping \(\alpha _*:} \rightarrow \Gamma _c^(M\leftarrow E_2)\) is Bastiani smooth. \(\square \)

Appendix C: Manifolds of Mappings and Sections

We begin with the definition of infinite-dimensional manifolds. As we mentioned earlier, we shall choose to model those on locally convex spaces in view of the results by [17, 18, 43].

Definition C.1

Let \(}\) be a Hausdorff topological space, we say that \(}\) admits a Bastiani smooth manifold structure if

(i):

there is a family \(\}_i, u_i, E_i)\}_\) where \(\}_i\}_\) is an open cover of \(}\), \(\_\) is a family of complete locally convex spaces and \(u_i:}_i \rightarrow E_i\) a family of homeomorphisms onto the open subsets \(u_i(}_i)\subseteq E_i\);

(ii):

for all \(i,j \in I\), having \(}_=}_i\cap }_j \ne \emptyset \), the mapping

$$\begin u_=u_j\circ u^_i: u_i(}_)\subseteq E_i \rightarrow u_j(}_)\subseteq E_j \end$$

and its inverse \(u_=u_i\circ u^_j\) are Bastiani smooth.

We then call charts elements of the family \(\}_i, u_i, E_i)\}_\).

It follows from condition (ii) above that the locally convex spaces \(E_i\) linearly isomorphic. A subset \(}\subset }\) of a differentiable manifold is called a splitting submanifold of \(}\) if for each \(p\in }\), there are charts \((},u,E)\) of \(}\) such that \(u(p)=0\in E\) and \(U(}\cap })=u(})\cap F\), where F is a closed vector subspace of E for which \(E=F\oplus F^c\). The collection of charts \(\}_i\cap }, u_i\vert _}_i\cap }},F_i)\}\) then makes \(}\) a manifold itself as per Definition C.1. A weaker notion of submanifolds requires that F is just a closed subspace of E, and in this case we say that \(}\) is a non-splitting submanifold.

Next we define the tangent bundle. Let \(}\) be a Bastiani smooth manifold with atlas \(\lbrace (}_,u_,E_)\rbrace \). A tangent vector is an equivalence class of elements \((p,v,U_,u_,E_)\), with \(p\in }\) and \(v \in E_\), where \((p,v,}_,u_,E_)\) and \((p',w,U_,u_,E_)\) are equivalent if \(p=p'\) and \(d(u_[u_(p)])(v)=w \). We denote by \(T_p}\) the set of all tangent vectors to p; moreover, setting \(T} = \bigsqcup _}} T_p} \) we obtain the space of tangent vectors of \(}\). It is easy to see that \(T}\) carries a natural structure of Bastiani smooth manifold. To wit, observe that we can always define a canonical projection \(\tau : T} \rightarrow } \). For the family of charts set \(\}}_, }_, E_ \times E_) \} \) where \(( }}_, _, E_) \) is a chart of \(}\), \(\widetilde}}_=\tau ^\big ( }_i\big )\), \(}_: \tau ^(}_i)\ni } \mapsto (u_(x),v) \in E_i\times E_i\).

The topology on \(T}\) is the unique one making each \(}_\) into a homeomorphism, and also, the transition mapping \(}_:(x,v) \mapsto (u_(y), du_[x](v))\) is Bastiani smooth since \(u_\) is itself smooth in the first place. It is easily shown that \(T}\) is Hausdorff, and thus, \(T}\) is a differentiable manifold according to Definition C.1.

Next we give a manifold structure to \(C^(M,N)\) with \(M, \ N\) smooth finite-dimensional manifolds. We first recall that given any Riemannian h on N there exists the Riemannian exponential \(\exp _y: U\subset T_yN \rightarrow N, w\mapsto \exp _y(w)\), where \(\exp _y(w)\) is the value of the geodesic starting at y with velocity w at time \(t=1\). Since \(\exp _y(0)=y\), and \(T_y\exp _y=id_\), \(\exp _y\) is a local diffeomorphism, then we can define a local diffeomorphism \((\tau _N,\exp ):}\subset TN \rightarrow }\subset N \times N : (y,w) \rightarrow (y, \exp _(w))\) onto an open subset \(}\) of the diagonal of \(N\times N\).

Theorem C.2

Let M, N be smooth finite-dimensional manifolds, then \(C^(M,N)\) is a Bastiani smooth manifold according to Definition C.1, modeled on the nuclear locally convex space \(\Gamma ^_c(M\leftarrow f^*TN)\).

Proof

Let \(\varphi \in C^(M,N)\), then define \(}_\) to be the subset of all \(g \in C^(M,N)\) with compact support with respect to \(\varphi \), such that \((\varphi ,\psi )(M)\subset (\tau _N,\exp )\big (}\big )\), then \(}_\) is an open subset, for example, by (vii) in Proposition B.10. Let then

$$\begin u_: }_ \ni \psi \mapsto u_(\psi )\in \Gamma ^_c(M\leftarrow \varphi ^*TN) \end$$

defined as follows:

$$\begin u_(\psi )(x)=(\tau _N,\exp )^ (\varphi (x),\psi (x))\simeq \big (\varphi (x),\exp ^_(\psi (x))\big ) . \end$$