Proof. Let \(\gamma\in BC(V,\mathbb{R})\) and \(p:\mathbb{R}\to\mathbb{R}\) be a continuous seminorm, then \[\lVert \gamma\circ \Theta|_W \lVert_{\infty,p}:=\sup_{x\in W} p(\gamma\circ \Theta|_W(x)) \leq \sup_{z\in V} p(\gamma(z)).\] Therefore \(\gamma\circ \Theta|_W \in BC(W,\mathbb{R})\). We define the continuous linear operator \[T:BC(\Theta|_W,\mathbb{R}):BC(V,\mathbb{R})\to BC(W,\mathbb{R}),\quad \gamma\mapsto \gamma\circ \Theta|_W\] with \(\lVert T\lVert_{op}\leq 1\). Hence, its graph \(\text{graph}(T)\) is closed in \(BC(V,\mathbb{R})\times BC(W,\mathbb{R})\). Since the inclusion map \(J:\mathcal{F}(U,\mathbb{R})\to BC(U,\mathbb{R})\) is continuous, we have \[\text{graph}(\mathcal{F}(\Theta|_W,\mathbb{R}))=(J\times J)^{-1}(\text{graph}(T)).\] Then \(\mathcal{F}(\Theta|_W,\mathbb{R})\) is continuous by the Closed Graph Theorem. ◻

Proof. As in the previous lemma, \(m_h\) is continuous since the operator \[M_h:BC(U,\mathbb{R})\to BC(U,\mathbb{R}),\quad \gamma\mapsto h\gamma\] is continuous linear, the graph of \(m_h\) is closed and therefore, \(m_h\) is continuous. ◻

Proof. Likewise to the previous lemmas, if \(BC_K(V,\mathbb{R}):=\{\gamma\in BC(V,\mathbb{R}): supp(\gamma)\subseteq K\}\) then the map \[BC (V,\mathbb{R})\to BC_K(U,\mathbb{R}),\quad \gamma \mapsto \tilde{\gamma}\] which extends functions by \(0\) is a linear isometry. ◻

Proof. Given \(\gamma_0\) in the domain \(D\) of \(\mathcal{F}(U/W,f)\), we have that \(\gamma_0(\overline{W})\) is a compact subset of \(V\). There exists a smooth function \(\chi\colon V\to\mathbb{R}\) with compact support \(K\subseteq V\) such that \(\chi(y)=1\) for all \(y\) in an open subset \(Y\subseteq V\) with \(\gamma_0(\overline{W})\subseteq Y\). Then \[g\colon E\to F,\quad g(y):=\left\{ \begin{array}{cl} \chi(y)f(y) & \,if \,y\in V;\\ 0 &\,if y\in E\setminus K \end{array}\right.\] is a smooth function. Since \(f|_Y=g|_Y\), we have that \[f\circ \gamma|_W=g\circ \gamma|_W\] for all \(\gamma\in D\) such that \(\gamma(\overline{W})\subseteq Y\), which is an open neighborhood of \(\gamma_0\) in \(D\). To see smoothness of \(\mathcal{F}(U/W,f)\) on some open neighborhood of \(\gamma_0\) (which suffices for the proof), we may therefore replace \(f\) with \(g\) and assume henceforth that \(V=E\), whence \(D\) is all of \(\mathcal{F}(U,E)\). It suffices to show that \(\mathcal{F}(U/W,f)\) is \(C^k\) for each \(k\in \mathbb{N}_0\), and we show this by induction. For the case \(k=0\), see Lemma 3.1.11. Let \(k\in \mathbb{N}_0\) now and assume that, for all \(E\), \(F\), \(U\), \(W\) and \(f\colon V\to F\) as in the lemma, with \(V=E\), the map \(\mathcal{F}(U/W,f)\) is \(C^k\). We claim that, for all \(\gamma,\eta\in \mathcal{F}(U,E)\), the directional derivative \[d\mathcal{F}(U/W,f)(\gamma,\eta)\] exists and equals \(\mathcal{F}(U/W, df)(\gamma,\eta)\), if we identify the locally convex spaces \(\mathcal{F}(U,E)\times\mathcal{F}(U,E)\) and \(\mathcal{F}(U,E\times E)\); thus \[\label{the-der-dir} d\mathcal{F}(U/W,f)(\gamma,\eta) =\mathcal{F}(U/W, df)(\gamma,\eta).\tag{1}\] If this is true, then \[d\mathcal{F}(U/W,f)=\mathcal{F}(U/W,df)\] is \(C^k\) by induction and thus continuous, showing that \(\mathcal{F}(U/W,f)\) is \(C^1\). Moreover, since \(\mathcal{F}(U/W,f)\) is \(C^1\) and \(d\mathcal{F}(U/W,f)=\mathcal{F}(U/W,df)\) is \(C^k\), the map \(\mathcal{F}(U/W,f)\) is \(C^{k+1}\), which completes the inductive proof. It only remains to prove the claim. To this end, let \(\gamma,\eta\in \mathcal{F}(U,E)\). Since \(\mathcal{F}(U/W,df)\) is continuous by the case \(k=0\), the map \[h\colon [0,1]\times [0,1] \to\mathcal{F}(W,F),\quad (t,s)\mapsto df\circ (\gamma+st\eta,\eta)|_W=\mathcal{F}(U/W,df)(\gamma+st\eta,\eta)\] is continuous. As \(\mathcal{F}(W,F)\) is assumed integral complete, for each \(t\in [0,1]\) the continuous path \(h(t,\cdot)\colon [0,1]\to \mathcal{F}(W,F)\) has a weak integral \[I(t):=\int_0^1 df\circ (\gamma+st\eta,\eta)|_W\, ds\] in \(\mathcal{F}(W,F)\). The function \(I\colon [0,1]\to\mathcal{F}(W,F)\) is continuous by the theorem on parameter-dependent integrals. For \(0\not= t\in [0,1]\), we consider the difference quotient \[\Delta(t)=\frac{\mathcal{F}(U/W,f)(\gamma+t\eta)-\mathcal{F}(U/W,f)(\gamma)}{t}= \frac{f\circ (\gamma+t\eta)|_W+f\circ\gamma|_W}{t}.\] Then \[\label{Delta-I} \Delta(t)=I(t).\tag{2}\] In fact, for each \(x\in W\) the point evaluation \[\varepsilon_x\colon \mathcal{F}(W,F)\to F,\quad\theta\mapsto \theta(x)\] is a continuous linear map. It therefore commutes with the weak integral and we obtain \[\begin{align} I(t)(x)&=&\varepsilon_x(I(t))=\int_0^1 \varepsilon_x(df\circ (\gamma+st\eta,\eta)|_W)\, ds\\ &=&\int_0^1 df(\gamma(x)+st\eta(x),\eta(x))\, ds =\frac{f(\gamma(x)+t\eta(x))-f(\gamma(x))}{t}\\ &=& \Delta(t)(x), \end{align}\] applying the mean value theorem to the smooth function \(f\). Thus (2 ) holds. Exploiting the continuity of \(I\), letting \(t\to 0\) we obtain \[\lim_{t\to 0}\Delta(t)=\lim_{t\to 0}I(t)=I(0) =\int_0^1 df\circ (\gamma,\eta)|_W\, ds=df\circ (\gamma,\eta)|_W,\] establishing (1 ). ◻

Proof. The map \(\Phi_\gamma\) is continuous by definition of the topology on \(\Gamma_\mathcal{F}(\gamma)\). We denote by \(S\) the vector space of elements \((\tau_i)_{i=1}^n\) such that \[\tau_{i}\circ \varphi_i(p) = d\phi_i \circ (T\phi_{j})^{-1} \Big(\phi_{j}\circ\gamma (p),\tau_j\circ \varphi_j(p)\Big)\] for all \(i,j\in\{1,..,k\}\) and \(p\in \varphi_i^{-1}(W_i)\cap \varphi_j^{-1}(W_j)\).
Clearly \(\text{Im}(\Phi_\gamma)\subseteq S\). Let \((\tau_i)_{i=1}^k \in S\), identifying the tangent bundle \(TV\) with \(V\times \mathbb{R}^n\) for any open subset \(V\subseteq \mathbb{R}^n\), we define the function \(\sigma:M\to TN\) by \[\sigma(p)=T\phi_i^{-1} \left( \phi_i\circ \gamma(p) ,\tau_i \circ \varphi_i(p)\right),\quad \text{if }p\in \varphi_i^{-1} (W_i)\text{ with }i\in \{1,...,k\}.\] We will show that the function \(\sigma\) is well defined. Let \(p\in \varphi_i^{-1}(W_i)\cap \varphi_j^{-1}(W_j)\), then \[\begin{align} T\phi_i^{-1} \left( \phi_i\circ \gamma(p) ,\tau_i \circ \varphi_i(p)\right) &= T\phi_j^{-1}\circ T\phi_j \circ T\phi_i^{-1} \left( \phi_i\circ \gamma(p) ,\tau_i \circ \varphi_i(p)\right) \\ &= T\phi_j^{-1}\left( \phi_j\circ \gamma(p), \tau_j\circ \varphi_j (p)\right) \end{align}\] Hence \(\sigma\) make sense. Moreover, we see that \(\pi_{N}\circ \sigma = \gamma\).
For \(\varphi_i|_{\varphi_i^{-1} (W_i)}: \varphi_i^{-1} (W_i) \to W_i\) and \(T\phi_i : TU_{\phi_i}\to V_{\phi_i}\times \mathbb{R}^n\) we have \[\begin{align} T\phi_i \circ \sigma \circ\varphi_i|_{\varphi_i^{-1} (W_i)} &= T\phi_i \circ \left( T\phi_i^{-1} \left( \phi_i\circ \gamma ,\tau_i \circ \varphi_i\right)\right) \circ\varphi_i|_{\varphi_i^{-1} (W_i)} \\ &= \left( \phi_i\circ\gamma\circ \varphi_i|_{\varphi_i^{-1} (W_i)} , \tau_i \right). \end{align}\] Since \(W_i\in \mathcal{U}\) we have \(\phi_i\circ \gamma\circ \varphi_i|_{\varphi_i^{-1} (W_i)} \in \mathcal{F}(W_i,V_{\phi_i})\) and \[T\phi_i \circ \sigma \circ \varphi_i|_{\varphi_i^{-1} (W_i)} \in \mathcal{F}(W_i,V_{\phi_i})\times \mathcal{F}(W_i,\mathbb{R}^n)\cong\mathcal{F}(W_i,V_{\phi_i}\times \mathbb{R}^n).\] Thus \(\sigma\in \Gamma_\mathcal{F}(\gamma)\). Evaluating we have \[\begin{align} \Phi_\gamma(\sigma) &= \left( d\phi_i \circ \sigma \circ \varphi_i^{-1}|_{W_i} \right)_{i=1}^k \\ &= \left( d\phi_i \circ T\phi_i^{-1} \left( \phi_i\circ \gamma ,\tau_i\circ \varphi_i \right)\circ \varphi_i^{-1}|_{W_i} \right)_{i=1}^k \\ &= \left( d\phi_i \circ T\phi_i^{-1} \left( \phi_i\circ \gamma\circ \varphi_i^{-1}|_{W_i} ,\tau_i \circ \varphi_i\circ \varphi_i^{-1}|_{W_i} \right) \right)_{i=1}^k \\ &= \left( d\phi_i \left( \gamma\circ \varphi_i^{-1}|_{W_i} ,d\phi_i^{-1} \circ \tau_i\right) \right)_{i=1}^k \\ &= \left( \tau_i \right)_{i=1}^k \end{align}\] Hence \(S\subseteq \text{Im}(\Phi_\gamma)\). The inverse of \(\Phi_\gamma\) is given by \(\Phi_\gamma^{-1} :\text{Im}(\Phi_\gamma)\to \Gamma_\mathcal{F}(\gamma)\) where \[\Phi_\gamma^{-1}\left((\tau_i)_{i=1}^k\right)(p)= T\phi_i^{-1} \left( \phi_i\circ \gamma(p) ,\tau_i \circ \varphi_i(p)\right),\quad \text{for }p\in \varphi_i^{-1}(W_i).\] Consider the arbitrary chart \(\alpha:U_\alpha\to V_\alpha\) of \(M\) with \(W_\alpha\in\mathcal{U}\) relatively compact in \(V_\alpha\) and the chart \(\beta:U_\beta\to V_\beta\) of \(N\) such that \(\alpha(U_\alpha)\subseteq U_\beta\). We will show that \[h_{\alpha,\beta}\circ \Phi_\gamma^{-1} :\text{Im}(\Phi_\gamma)\to \mathcal{F}(W, \mathbb{R}^n),\quad \tau=(\tau_i)_{i=1}^k\mapsto d\beta\circ \sigma_\tau\circ \alpha^{-1}|_W\] is continuous. Since \(M=\cup_{i=1}^k \varphi_i^{-1}(W_i)\), we define the open set \[Q_j:=\alpha\left(U_\alpha\cap \varphi_j^{-1}(W_j)\right).\] Without loss of generality, we assume that there exists \(r\in \{1,..,k\}\) such that \(Q_j\neq \emptyset\) for each \(j\in\{1,...,r\}\) and \(Q_j=\emptyset\) for each \(j\in\{r+1,..,k\}\).
Then, for \(\tau \in \text{Im}(\Phi_\gamma)\) and \(j\in\{1,...,r\}\) we have \[\begin{align} d\beta \circ \sigma_\tau \circ \alpha^{-1}|_{Q_j} &= d\beta \circ T\phi_j^{-1} \left( \phi_j\circ \gamma ,\tau_j \circ \varphi_j\right) \circ \alpha^{-1}|_{Q_j} \\ &= \mathcal{F}( W_j,d\beta \circ T\phi_j^{-1})\circ \left(\phi_i\circ \gamma\circ \alpha^{-1}|_{Q_i}, \mathcal{F}(\varphi_j \circ \alpha^{-1}|_{Q_j}, \mathbb{R}^n) \circ \tau_j\right). \end{align}\] Hence \(d\beta \circ \sigma_\tau \circ \alpha^{-1}|_{Q_j} \in \mathcal{F}_{\text{loc}} (Q_i,\mathbb{R}^n)\). This enables us to define the continuous map \[\Lambda: \text{Im}(\Phi_\gamma)\to \prod_{i=1}^r \mathcal{F}_{\text{loc}} (Q_i,\mathbb{R}^n). \quad \tau \mapsto \left(d\beta\circ \sigma_\tau \circ \alpha^{-1}|_{V_i}\right)_{i=1}^r\] where the image set \(\text{Im}(\Lambda)\) coincides with the subspace \[\left\{ (\beta_1,...,\beta_r)\in \prod_{i=1}^r \mathcal{F}_{\text{loc}} (Q_i,\mathbb{R}^n): \left(\forall i,j\in\{1,...,r\}\right) \beta_i|_{Q_i\cap Q_j} = \beta_j|_{Q_i\cap Q_j} \right\}.\] For \((\beta_1,...,\beta_r)\in \prod_{i=1}^r \mathcal{F}_{\text{loc}} (Q_i,\mathbb{R}^n)\), we denote the gluing function \[\beta(x):=\beta_j(x),\quad \text{if } x\in Q_i .\] For each \(W\in \mathcal{U}\) relatively compact in \(Q_1\cup...\cup Q_r\) we have \(\beta|_W\in\mathcal{F}(W,\mathbb{R}^n)\) and the map \[\text{glue}_W:\text{Im}(\Lambda)\to \mathcal{F}(W,\mathbb{R}^n),\quad (\beta_1,...,\beta_r)\mapsto \beta|_W\] is continuous [1]. Therefore \(h_{\varphi,\phi}\circ \Phi_\gamma^{-1}\) is continuous since \[h_{\varphi,\phi}\circ \Phi_\gamma^{-1} = \text{glue}_W\circ \Lambda.\] Hence \(\Phi_\gamma^{-1}\) is continuous. ◻

Proof. Since \(f\) and \(Tf\) are smooth, by Lemma 5 we have \(f\circ \gamma \in \mathcal{F}(M,N_2)\) and \[Tf\circ \sigma \in \mathcal{F}(M,TN_2),\quad \text{ for all }\sigma\in \Gamma_\mathcal{F}(\gamma).\] Since \(\sigma(t)\in T_{\gamma (t)}N_1\) for each \(t\in [a,b]\), we have \(T_{\gamma(t)}f \circ \sigma(t)\in T_{f\circ \gamma(t)} N_2\), thus \[\pi_{TN_2}\circ (Tf\circ \sigma) = f\circ \gamma\] and \(\widetilde{f}\) is well defined.
Let \(\gamma\in \mathcal{F}(M,N_1)\). For \(i\in\{1,...,k\}\), let \(\varphi_{M,i}:U_{M,i}\to V_{M,i}\) be charts of \(M\) such that there exists \(W_{M,i}\in \mathcal{U}\) relatively compact in \(V_{M,i}\) with \(M=\cup_{i=1}^k \varphi_{M,i}^{-1}(W_{M,i})\) and there exist chart \(\phi_{1,i} : U_{\phi_{1,i}}\to V_{\phi_{1,i}}\) and \(\phi_{2,i} : U_{\phi_{2,i}}\to V_{\phi_{2,i}}\) of \(N_1\) and \(N_2\) respectively, such that \(\gamma\left(U_{M,i}\right)\subseteq U_{\phi_{1,i}}\) and \(f\left(U_{\phi_{1,i}}\right)\subseteq U_{\phi_{2,i}}\). We may assume that \(V_{\phi_{1,i}}=\mathbb{R}^{n_1}\) and \(V_{\phi_{2,i}}=\mathbb{R}^{n_2}\).
Let \(\gamma_i:=\phi_{1,i}\circ \gamma|_{W_{M,i}}\) and \(W_{M,i}^\prime \in \mathcal{U}\) be relatively compact in \(U_{M,i}\), containing the closure of \(W_{M,i}\). For \(i\in\{1,...,k\}\) we define the smooth map \[f_i:=d\phi_{2,i} \circ Tf \circ T\phi_{1,i}^{-1}: TV_{\psi_{1,i}}\subseteq \mathbb{R}^{2n_1} \to \mathbb{R}^{n_2}\] and \[G: \prod_{i=1}^k \mathcal{F}(W_{M,i}^\prime,\mathbb{R}^{2n_1}) \to \prod_{i=1}^k \mathcal{F}(W_{M,i},\mathbb{R}^{n_2}), \quad (\xi_i)_{i=1}^k\mapsto \left(f_i \circ (\gamma_i,\xi_i|_{W_{M,i}})\right)_{i=1}^k\] which is continuous by Lemma 5. We consider the linear topological embeddings \[\Phi_\gamma:\Gamma_{\mathcal{F}} (\gamma) \to \prod_{i=1}^k \mathcal{F}(W_{M,i},\mathbb{R}^{n_1}) ,\quad \sigma \mapsto \left( d\phi_{1,i} \circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}} \right)_{i=1}^k\] and \[\Phi_{f\circ\gamma}:\Gamma_{\mathcal{F}} (f\circ\gamma) \to \prod_{i=1}^k \mathcal{F}(W_{M,i},\mathbb{R}^{n_2}) ,\quad \tau \mapsto \left( d\phi_{i,2} \circ \tau \circ \varphi_{M,i}^{-1}|_{W_{M,i}} \right)_{i=1}^k.\] Then \[G(\text{Im}(\Phi_\gamma))\subseteq \text{Im}(\Phi_{f\circ \gamma}).\] Indeed, consider \(\sigma\in \Gamma_\mathcal{F}(\eta)\). Then for each \(i\in\{1,...,k\}\) \[\tau_i:=f_i(T\phi_{1,i} \circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}}) = d\phi_{2,i} \circ Tf \circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}}.\] And for all \(i,j\in\{1,..,k\}\) and \(p\in \varphi_{M,i}^{-1}(W_{M,i})\cap \varphi_{M,j}^{-1}(W_{M,j})\), we have \[\begin{align} \tau_i \circ \varphi_{M,i} (p) &= d\phi_{2,i} \circ Tf\circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}}\circ \varphi_{M,i}(p) \\ &= d\phi_{2,i} \circ Tf\circ \sigma (p) \\ &= d\phi_{2,i} \Big(f\circ \gamma (p),df \circ \sigma (p)\Big)\\ &= d\phi_{2,i} \circ (T\phi_{2,j})^{-1} \Big(\phi_{2,j}\circ f\circ\gamma (p),d\phi_{2,j} \circ df \circ \sigma\circ \varphi_{M,j}^{-1}|_{W_{M,j}}\circ \varphi_{M,j}(p) \Big)\\ &= d\phi_{2,i} \circ (T\phi_{2,j})^{-1} \Big(\phi_{2,j}\circ f \circ\gamma (p),\tau_j\circ \varphi_{M,j}(p)\Big) \end{align}\] Hence \[\left(f_i \circ \left(d\phi_{1,i}\circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}}\right)\right)_{i=1}^k \in \text{Im}(\Phi_{f\circ \gamma}).\] In consequence \[\widetilde{f} = \Phi_{f\circ\gamma}^{-1} \circ G \circ \Phi_{\gamma}.\] Thus \(\widetilde{f}\) is continuous and the linearity is clear. ◻

Proof. Let \(f:=(\pi_1,\pi_2)\), by Proposition 18 the map \(\mathcal{P}\) is continuous and clearly linear. For \(j\in\{1,2\}\) and \(i\in\{1,...,k\}\), let \(\varphi_{j,i}:U_{j,i}\to V_{j,i}\) be charts of \(M\) such that there exists \(W_{j,i}\in \mathcal{U}\) relatively compact in \(V_{j,i}\) with \(M=\cup_{i=1}^k \varphi_{j,i}^{-1}(W_{j,i})\) and there exists a chart \(\phi_{j,i} : U_{\phi_{j,i}}\to V_{\phi_{j,i}}\) such that \(\gamma_j\left(U_{j,i}\right)\subseteq U_{\phi_{j,i}}\). Then we have the homeomorphisms \[\Phi_{\gamma_j}:\Gamma_\mathcal{F}(\gamma_j)\to \text{Im}(\Phi_{\gamma_j}).\] We consider the isomorphism of topological vector spaces \[\alpha :\prod_{i=1}^k\mathcal{F}(W_{j,i},\mathbb{R}^{n_1})\times \mathcal{F}(W_{j,i},\mathbb{R}^{n_2}) \to \prod_{i=1}^k \mathcal{F}(W_{j,i},\mathbb{R}^{n_1}\times \mathbb{R}^{n_2}), \quad (\xi_1,\xi_2)\mapsto \alpha(\xi_1,\xi_2).\] If \(\Phi_\gamma\) denotes the homeomorphism for \(\gamma:=(\gamma_1,\gamma_2)\in \mathcal{F}(M,N_1\times N_2)\) as Proposition 15, then \[\Theta:=\Phi_\gamma^{-1}\circ \alpha \circ \left(\Phi_{\gamma_1}\times \Phi_{\gamma_2}\right)\] is continuous linear and for each \(\sigma\in \Gamma_\mathcal{F}(\gamma_1)\) and \(\tau\in \Gamma_\mathcal{F}(\gamma_2)\) we have \[\mathcal{P}\left(\Theta(\sigma,\tau)\right)=(\sigma,\tau).\] Hence \(\mathcal{P}\) is surjective and thus bijective, with \(\mathcal{P}^{-1} = \Theta\) a continuous map. ◻

Proof. Let \(\gamma\in\mathcal{F}(M_2,N)\). Let \(\phi_1:U_1 \to V_1\) and \(\phi_2:U_2 \to V_2\) charts of \(M_1\) and \(M_2\) respectively such that \(\Theta(U_1)\subseteq U_2\). If \(\phi_N:U_N \to V_N\) is a chart of \(N\) such that \((\gamma\circ \Theta)(U_1)\subseteq U_N\) then \[\phi_N \circ \left( \gamma \circ \Theta \right) \circ \phi_1^{-1} = \phi_N \circ \gamma \circ \phi_2^{-1}\circ \phi_2 \circ \Theta\circ \phi_1^{-1}.\] Since \(\zeta:=\phi_N \circ \gamma \circ \phi_2^{-1} \in \mathcal{F}_{\text{loc}}(V_\psi,\mathbb{R}^n)\) and the map \(g:=\phi_2 \circ \Theta\circ \phi_1^{-1}:V_\varphi \to V_\psi\) is a smooth diffeomorphism, by Lemma 9 we have that \(\zeta\circ g \in \mathcal{F}_{\text{loc}}(V_\varphi,\mathbb{R}^n)\). Thus \[\gamma \circ \Theta \in \mathcal{F}(M_1,N).\] Analogously, we can show that \(\sigma\circ \Theta\in \Gamma_{\mathcal{F}} (\gamma\circ \Theta)\) for each \(\sigma \in \Gamma_\mathcal{F}(\eta)\).
By compactness of \(M_1\) and \(M_2\), for \(i\in\{1,...,k\}\) we consider charts \(\phi_{1,i}:U_{1,i}\to V_{1,i}\) of \(M_1\) such that there exists \(W_{1,i}\in \mathcal{U}\) relatively compact in \(V_{1,i}\) with \(M_1=\cup_{i=1}^k \phi_{1,i}^{-1}(W_{1,i})\) and charts \(\phi_{2,i}:U_{2,i}\to V_{2,i}\) of \(M_2\) such that there exists \(W_{2,i}\in \mathcal{U}\) relatively compact in \(V_{2,i}\) with \(M_2=\cup_{i=1}^k \phi_{2,i}^{-1}(W_{2,i})\) such that there exists a chart \(\phi_{N,i} : U_{N.i}\to V_{N,i}\) of \(N\) such that \(\Theta(W_{1,i})\subseteq W_{2,i}\) and \(\gamma\left(U_{2,i}\right)\subseteq U_{N,i}\). We define the topological embeddings \[\Phi_\gamma:\Gamma_{\mathcal{F}} (\gamma) \to \prod_{i=1}^k \mathcal{F}(W_{2,i},\mathbb{R}^n) ,\quad \sigma \mapsto \left( d\phi_{N,i} \circ \sigma \circ \phi_{2.i}^{-1}|_{W_{2,i}} \right)_{i=1}^k\] and \[\Phi_{\gamma\circ \Theta}:\Gamma_{\mathcal{F}} (\gamma\circ \Theta) \to \prod_{i=1}^k \mathcal{F}(W_{1,i},\mathbb{R}^n) ,\quad \sigma \mapsto \left( d\phi_{N,i} \circ \sigma \circ \Theta \circ \phi_{1,i}^{-1}|_{W_{1,i}} \right)_{i=1}^k\] Since the map \[\Theta_i:=\phi_{2.i}\circ \Theta\circ \phi_{1,i}^{-1}|_{W_{1,i}}:W_{1,i}\to \Theta_i(W_{2,i})\] is a smooth diffeomorphism, the map \[\mathcal{F}(\Theta_i,\mathbb{R}^n): \mathcal{F}_{\text{loc}}(\Theta_i(W_{2,i}),\mathbb{R}^n)\to \mathcal{F}_{\text{loc}}(W_{1,i},\mathbb{R}^n), \quad \tau\mapsto \tau \circ \Theta_i\] and thus \[\overline{\Theta}: \prod_{i=1}^k\mathcal{F}_{\text{loc}}(\Theta_i(W_{2,i}),\mathbb{R}^n)\to \prod_{i=1}^k\mathcal{F}_{\text{loc}}(W_{2,i},\mathbb{R}^n), \quad (\tau_i)_{i=1}^n \mapsto \left(\tau_i \circ \Theta_i\right)_{i=1}^k\] are continuous. We will show that \(\overline{\Theta}(\text{Im}(\Phi_\gamma))\subseteq \text{Im}(\Phi_{\gamma\circ \Theta})\).
For each \(i, j\in\{1,...,k\}\) and \(\sigma\in \Gamma_\mathcal{F}(\eta)\), if \[\begin{align} \tau_i &:= d\phi_{N,i} \circ \sigma \circ \phi_{2,i}^{-1}|_{W_{2,i}}\circ \Theta_i \\ &= d\phi_{N,i} \circ \sigma \circ \Theta\circ \phi_{1,i}^{-1}|_{W_{1,i}} \end{align}\] then \[\begin{align} \tau_{i}\circ \phi_{1,i}(p) &= d\phi_{N,i}\circ \sigma \circ \Theta\circ \phi_{1,i}^{-1}\circ \phi_{1,i} (p) \\ &= d\phi_{N,i}\circ \sigma \circ \Theta (p)\\ &= d\phi_{N,i} \circ (T\phi_{N,j})^{-1} \Big(\phi_{N,j}\circ\gamma (p),d\phi_{N,j} \circ \sigma \circ \Theta(p)\Big) \\ &= d\phi_{N,i} \circ (T\phi_{N,j})^{-1} \Big(\phi_{N,j}\circ\gamma (p),\tau_j\circ \phi_{1,j}(p)\Big) \end{align}\] Hence \(\overline{\Theta}(\text{Im}(\Phi_\gamma))\subseteq \text{Im}(\Phi_{\gamma\circ \Theta})\). In consequence, since \[L_\Theta = \Phi_{\gamma\circ \Theta}^{-1} \circ \left( \mathcal{F}(\Theta_i,\mathbb{R}^n) \right)_{i=1}^k \circ \Phi_\gamma\] the map \(L_\Theta\) is continuous. ◻

Proof. Since the evaluation map \[\tilde{\epsilon}: \Gamma_C(\gamma)\times M\to TN, \quad (\sigma, p)\mapsto \sigma(p)\] is continuous and the evaluation map \(\tilde{\epsilon}_p : \Gamma_C (\gamma)\to TN\), \(\sigma\mapsto \sigma(p)\) is smooth for each \(p\in M\) (see [3]). Then \(\epsilon = \tilde{\epsilon} \circ (J_\Gamma,\text{Id}_\mathbb{R})\) and \(\epsilon_p = \tilde{\epsilon}_p \circ J_\Gamma\), where \(J_\Gamma:\Gamma_\mathcal{F}(\gamma)\to \Gamma_C(\gamma)\) is the inclusion map, which is smooth by Remark 20. ◻

Proof. We endow \(\mathcal{F}(M,N)\) with the final topology with respect to the family \(\Psi_\gamma\colon {\mathcal{V}}_\gamma\to {\mathcal{U}}_\gamma\) for each \(\gamma\in \mathcal{F}(M,N)\). If we define the maps \(\Psi_\gamma^C\colon {\mathcal{V}}_\gamma^C\to {\mathcal{U}}_\gamma^C\) on the space of continuous functions \(C(M,N)\) for each \(\gamma\in C(M,N)\) then the final topology on \(C(M,N)\) coincides with its topology (the compact-open topology), whence the inclusion map \[J: \mathcal{F}(M,N) \to C(M,N), \quad \gamma \mapsto \gamma\] is continuous. Moreover, since \[\mathcal{U}_{J(\gamma)}^C := \{ \xi \in C(M,N) : (J(\gamma),\xi)(M) \subseteq \Omega' \}\] is open in \(C(M,N)\), the set \[\mathcal{U}_\gamma = \mathcal{U}_\gamma^C \cap \mathcal{F}(M,N)\] is open in \(\mathcal{F}(M,N)\).
The goal is to make to the family \(\{ (\mathcal{U}_\gamma, \Psi_\gamma^{-1}) : \gamma \in \mathcal{F}(M,N) \}\) an atlas for the manifold structure.
Let \(\gamma,\xi\in \mathcal{F}(M,N)\), it remains to show that the charts are compatible, i.e. the smoothness of the map \[\label{Fcompatiblecharts} \Lambda_{\xi,\gamma}:= \Psi_\xi^{-1} \circ \Psi_\gamma : \Psi_\gamma^{-1}(\mathcal{U}_\gamma \cap \mathcal{U}_\xi)\subseteq \Gamma_{\mathcal{F}} (\gamma) \to \Gamma_{\mathcal{F}} (\xi) , \quad \sigma \mapsto \theta_N^{-1} \circ (\xi, \Sigma \circ \sigma ).\tag{3}\] For \(i\in\{1,...,k\}\), let \(\varphi_i:U_{M,i}\to V_{M,i}\) be charts of \(M\) and \(W_{M,i}\in \mathcal{U}\) such that \(W_{M,i}\) is relatively compact in \(V_i\) with \(M=\cup_{i=1}^k \varphi_i^{-1}(W_{M,i})\) and charts \(\phi_i^\gamma : U_{N,i}^\gamma\to V_{N,i}^\gamma\) and \(\phi_i^\xi : U_{N,i}^\xi\to V_{N,i}^\xi\) of \(N\) such that \(\gamma\left( U_{M,i})\right)\subseteq U_{N,i}^\gamma\) and \(\xi\left(U_{M,i}\right)\subseteq U_{N,i}^\xi\).
We will study the smoothness of the composition \[\Phi_\xi \circ\Lambda_{\xi,\gamma} : \Psi_\gamma^{-1}(\mathcal{U}_\gamma \cap \mathcal{U}_\xi) \to \text{Im}(\Phi_\xi )\subseteq\prod_{i=1}^k \mathcal{F}(W_{M,i},\mathbb{R}^n),\quad \sigma \mapsto \left( d\phi_i^\xi \circ \Lambda_{\xi,\gamma}(\sigma) \circ \varphi_i^{-1}|_{W_{M,i}} \right)_{i=1}^k\] which is equivalent to the smoothness of \(\Lambda_{\xi,\gamma}\), where \(\Phi_\xi\) is the linear topological embedding as in Proposition 15. By Definition 5 c), we find \(W_M^\prime\) in \(\mathcal{U}\) such that \(\overline{W_M^\prime}\) is relatively compact in \(V_i\) and \(W_{M,i}^\prime\) contains the closure of \(W_{M,i}\).
For each \(i\in\{1,...,k\}\) and \(\sigma \in \Psi_\gamma^{-1}(\mathcal{U}_\gamma \cap \mathcal{U}_\xi)\) we have \[d\phi_i^\xi \circ \left(\Psi_\xi^{-1}( \Psi_\gamma (\sigma))\right) \circ \varphi_i^{-1}|_{W_{M,i}^\prime} = d\phi_i^\xi \circ \theta_N^{-1} \circ \left(\xi\circ \varphi_i^{-1}|_{W_{M,i}^\prime}, \Sigma \circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime}\right).\] Since \(\sigma\left(\varphi_i^{-1}(W_{M,i}^\prime)\right) \subseteq TU_{\phi_i^\gamma}\) we can do \[\begin{align} \Sigma \circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime} &= \Sigma\circ \left(T\phi_i^\gamma\right)^{-1} \circ T\phi_i^\gamma \circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime} \\ &=\Sigma\circ \left(T\phi_i^\gamma\right)^{-1}\left(\phi_i^\gamma \circ \gamma\circ\varphi_i^{-1}|_{W_{M,i}^\prime} , d\phi_i^\gamma\circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime}\right) \end{align}\] and \[\xi\circ \varphi_i^{-1}|_{W_{M,i}^\prime} = \left(\phi_i^\xi\right)^{-1} \circ\left( \phi_i^\xi\circ \xi\circ \varphi_i^{-1}|_{W_{M,i}^\prime}\right).\] Because all of the functions involved are continuous and have an open domain, also the composition \[H_i(x,y,z) := d\phi_i^\xi \circ \theta_N^{-1} \circ \left(\left(\phi_i^\xi\right)^{-1}\left( x \right), \Sigma\circ \left(T\phi_i^\gamma\right)^{-1} \left( y , z \right) \right),\] has an open domain \(\mathcal{O}_i\). Hence the map \(H_i: \mathcal{O}_i \to E\) is smooth.
By Lemma 6, the map \[h_i:=\mathcal{F}(W_{M,i}^\prime/ W_{M,i}, H_i)\] is smooth. By the preceding \[\Phi_\xi\circ\Lambda_{\xi,\gamma} = h_i(\phi_i^\xi \circ \xi\circ \varphi_i^{-1}|_{W_{M,i}^\prime}, \phi_i^\gamma\circ \gamma\circ \varphi_i^{-1}|_{W_{M,i}^\prime}, d\phi_i^\gamma\circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime}),\] which is a smooth function of \(\sigma\), using that the maps

\[\Gamma_\mathcal{F}(\gamma)\to \mathcal{F}(W_{M,i}^\prime,\mathbb{R}^n),\quad \sigma\mapsto d\phi_i^\gamma\circ \sigma\circ \varphi_i^{-1}|_{W_{M,i}^\prime}\] are continuous linear by definition of the topology of \(\Gamma_\mathcal{F}(\gamma)\). Therefore \(\Psi_\xi^{-1} \circ \Psi_\gamma\) is smooth. ◻

Proof. The map is well defined by Lemma 12. Let \((\Omega_1,\Sigma_1)\) and \((\Omega_2,\Sigma_2)\) be the local addition for \(N_1\) and \(N_2\) respectively. We consider the charts \((\mathcal{U}_\gamma, \Psi_\gamma^{-1})\) and \((\mathcal{U}_{f\circ\gamma}, \Psi_{f\circ\gamma}^{-1})\) in \(\gamma \in \mathcal{F}(M,N)\) and \(f\circ \gamma \in \mathcal{F}(M,N)\) respectively. We define \[F(\sigma):=\Psi_{f\circ\gamma}^{-1} \circ \mathcal{F}(M,f) \circ \Psi_\gamma (\sigma) = (\pi_{N},\Sigma_2)^{-1} \circ \Big(f\circ \gamma , f\circ \Sigma_1 \circ \sigma \Big)\] for all \(\sigma \in \Psi_\gamma^{-1}\left( \mathcal{U}_\gamma \cap \mathcal{F}(M,f)^{-1}(\mathcal{U}_{f\circ\gamma})\right)\).
We will proceed as in the proof of the Theorem 27. For \(i\in\{1,...,k\}\), let \(\varphi_i:U_{M,i}\to V_{M,i}\) be charts of \(M\), \(W_{M,i}\in \mathcal{U}\) such that \(W_{M,i}\) is relatively compact in \(V_i\) with \(M=\cup_{i=1}^k \varphi_i^{-1}(W_{M,i})\) and \(\phi_{1,i} : U_{1,i}\to V_{1,i}\) and \(\phi_{2,i} : U_{2,i}\to V_{2,i}\) charts of \(N_1\) and \(N_2\) respectively such that \(\gamma\left( U_{M,i} \right)\subseteq U_{1,i}\) and \((f\circ \gamma) (U_{M,i})\subseteq U_{2,i}\). We will study the smoothness of the composition \[\Phi_{f\circ\xi} \circ F : \Psi_\xi^{-1}\left( \mathcal{U}_\gamma \cap \mathcal{F}(M,f)^{-1}(\mathcal{U}_{f\circ\gamma})\right) \to \text{Im}(\Phi_{f\circ\xi}),\quad \sigma \mapsto \left( d\phi_{2,i} \circ F(\sigma) \circ \varphi_{M,i}^{-1}|_{W_{M,i}} \right)_{i=1}^k\] where \(\Phi_{f\circ\xi}\) is the linear topological embedding as in Proposition 15. Using Definition 5 c), we find sets \(W_{M,i}^\prime\) in \(\mathcal{U}\) which are relatively compact in \(U_{M,i}\) and contain \(W_{M,i}\). For each \(i\in\{1,...,k\}\) and \(\sigma \in \Psi_\gamma^{-1}\left( \mathcal{U}_\gamma \cap \mathcal{F}(M,f)^{-1}(\mathcal{U}_{f\circ\gamma})\right)\) we have \[\begin{align} d\phi_{2,i} \circ F(\sigma) \circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} &= d\phi_{2,i} \circ(\pi_{N},\Sigma_2)^{-1} \circ \Big(f\circ \gamma , f\circ \Sigma_1 \circ \sigma \Big) \circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} \\ &= d\phi_{2,i} \circ(\pi_{N},\Sigma_2)^{-1} \circ \Big(f\circ \gamma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} , f\circ \Sigma_1 \circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime}\Big). \end{align}\] Since \((f\circ \gamma) (U_{M,i})\subseteq U_{2,i}\) we have \[f\circ \gamma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} = \phi_{2,i}^{-1}\circ \left( \phi_{2,i} \circ f\circ \gamma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime } \right).\] And since \(\gamma\left( \varphi_{M,i}^{-1}(V_{M,i})\right)\subseteq U_{1,i}\) we have \(\sigma\left(\varphi_{M,i}^{-1}(W_{M,i})\right) \subseteq TU_{1,i}\) whence \[\begin{align} f\circ\Sigma_1 \circ \sigma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} &= f\circ\Sigma_1\circ T\phi_{1,i}^{-1} \circ T\phi_{1,i} \circ \sigma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime} \\ &=f\circ\Sigma_1\circ T\phi_{1,i}^{-1}\left(\phi_{1,i} \circ \gamma\circ\varphi_{M,i}^{-1}|_{W_{M,i}^\prime} , d\phi_{1,i}\circ \sigma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime}\right). \end{align}\] Let \[H_i(x,y,z):= d\phi_{2,i} \circ (\pi_{N},\Sigma_2)^{-1} \circ (\phi_{2,i}^{-1} \left( x \right),f\circ \Sigma_1\circ T\phi_{1,i}^{-1} \left( y , z \right) ).\] Then \(H_i\) is defined on an open subset of \(\mathbb{R}^{n_2}\times \mathbb{R}^{n_1}\times \mathbb{R}^{n_1}\) and the \(\mathbb{R}^{n_2}\)-valued function \(H_i\) so obtained is smooth (because it is a composition of smooth functions).
By Lemma 6, also the corresponding mappings \[h_i:=\mathcal{F}(W_{M,i}^\prime / W_{M,i}, H_i)\] between functions spaces are smooth. By the above, we have \[(\Phi_{f\circ \xi}\circ F)(\sigma) h_i\left(\phi_{2,i}\circ f\circ \gamma \circ \varphi_{M,i}^{-1}|_{W_{M,i}^\prime}, \phi_{1,i}\circ \gamma\circ \varphi_{M,i}^{-1}|_{W_{M,i}^{-1}}, d\phi_{1,i}\circ \sigma \circ \varphi_{M,i}^{-1}|_{W_{M,i}^{-1}} \right)\] which is a smooth function of \(\sigma\), using that \[\Gamma_\mathcal{F}(\gamma)\to \mathcal{F}(W_{M,i},\mathbb{R}^n),\quad \sigma\mapsto d\phi_{1,i}\circ \sigma\circ \varphi_{M,i}^{-1}|_{W_{M,i}}\] is a continuous linear map by definition. Therefore \(\mathcal{F}(M,f)\) is smooth. ◻

Proof. If \((\Omega_1,\Sigma_1)\) and \((\Omega_1,\Sigma_1)\) are the local additions on \(N_1\) and \(N_2\) respectively, then we can assume that the local addition on \(N_1\times N_2\) is \[\Sigma:=\Sigma_1\times\Sigma_2: \Omega_1\times \Omega_2 \to N_1\times N_2\] where \(\Omega_1\times \Omega_2 \subseteq TN_1 \times TN_2 \cong T(N_1\times N_2)\). The map \(\mathcal{P}\) is smooth as consequence of the smoothness of the maps \[\mathcal{F}(M,\text{pr}_j):\mathcal{F}(M,N_1 \times N_2)\to \mathcal{F}(M,N_i),\] for each \({i\in\{1,2\}}\) by the previous results.
Let \((\mathcal{U}_{\gamma}\times \mathcal{U}_{\gamma}, \Psi_{\gamma_1}^{-1}\times \Psi_{\gamma_2}^{-1})\) and \((\mathcal{U}_\gamma, \Psi_\gamma^{-1})\) be charts in \((\gamma_1,\gamma_2) \in \mathcal{F}(M,N_1)\times \mathcal{F}(M,N_2)\) and \(\mathcal{P}^{-1}(\gamma_1,\gamma_2)=\gamma \in \mathcal{F}(M,N_1\times N_2)\) respectively. Since the map \[\mathcal{Q}: \Gamma_{\mathcal{F}}(\gamma)\to \Gamma_{\mathcal{F}}(\gamma)\times \Gamma_{\mathcal{F}}(\gamma_2),\quad \tau \mapsto (\text{q}_1,\text{q}_2)\circ \tau\] where \(\text{q}_1\) and \(\text{q}_2\) are the corresponding projection of the space, is an diffeomorphism of vector spaces (by Lemma 3.3 and Proposition 15), we have \[\begin{align} \Psi_\gamma^{-1} \circ \mathcal{P}^{-1}\circ (\Psi_{\gamma_1}\times \Psi_{\gamma_2}) (\sigma_1,\sigma_2) &= (\pi_{N_1\times N_2}, \Sigma)^{-1}\circ \left( \gamma, \mathcal{P}^{-1} \circ (\Sigma_1\times \Sigma_2)(\sigma_1,\sigma_2)\right)\\ &= (\pi_{N_1\times N_2}, \Sigma)^{-1}\circ \left( \gamma, \Sigma\circ \mathcal{Q}^{-1}(\sigma_1,\sigma_2) \right)\\ &= \mathcal{Q}^{-1}(\sigma_1,\sigma_2) \end{align}\] for all \((\sigma_1,\sigma_2) \in (\Psi_{\gamma_1}^{-1}\times \Psi_{\gamma_2}^{-1}) \left( \mathcal{U}_{\gamma_1}\times \mathcal{U}_{\gamma_2} \cap \mathcal{P}(\mathcal{U}_{\gamma})\right)\). Hence \(\mathcal{P}^{-1}\) is smooth. ◻

Proof. By Proposition 22 we know that the map is well defined. Let \((\mathcal{U}_\gamma, \Psi_\gamma^{-1})\) and \((\mathcal{U}_{\gamma\circ \Theta}, \Psi_{\gamma\circ \Theta}^{-1})\) be charts in \(\gamma \in \mathcal{F}(M_2,N)\) and \(\gamma\circ \Theta \in\mathcal{F}(M_1,N)\) respectively, then we have \[\Psi_{\gamma\circ \Theta}^{-1} \circ \mathcal{F}(\Theta,N) \circ \Psi_\gamma (\sigma) = \theta_N^{-1}\circ (\gamma\circ \Theta, \Sigma \circ (\sigma\circ \Theta ))\] for all \(\sigma \in \Psi_\gamma^{-1}\left( \mathcal{U}_\gamma \cap \mathcal{F}(\Theta,N)^{-1}(\mathcal{U}_{\gamma\circ \Theta})\right)\).
Let \(\alpha=\gamma\circ \Theta :M_1\to N\) and \(\tau=\sigma\circ \Theta:M_2\to TN\), then \(\tau \in \Gamma_{\mathcal{F}}(\alpha)\) and \[\begin{align} \Psi_{\gamma\circ \Theta}^{-1} \circ \mathcal{F}(\Theta,N)\circ \Psi_\gamma (\sigma) &= \theta_N^{-1}\circ (\alpha, \Sigma \circ \tau ) \\ &= \Psi_{\alpha}^{-1}\circ \Psi_\alpha (\tau) \\ &= \tau \\ &= \sigma \circ \Theta. \end{align}\] Hence, \(\Psi_{\gamma\circ \Theta}^{-1} \circ\mathcal{F}(\Theta,N) \circ \Psi_\gamma\) is a restriction of the map \[L_\Theta:{\Gamma_{\mathcal{F}}(\gamma)}\to {\Gamma_{\mathcal{F}}(\gamma\circ \Theta)},\quad \sigma\to \sigma\circ \Theta\] which is linear and continuous by Proposition 22. ◻

Proof. Define the smooth map \[C_\gamma: \mathcal{F}(M,K) \to \mathcal{F}(M,L) \times \mathcal{F}(M,K), \quad \xi\mapsto (\gamma,\xi).\] Identifying \(\mathcal{F}(M,L) \times \mathcal{F}(M,K)\) with \(\mathcal{F}(M,L\times K)\), we have \[f_* = \mathcal{F}(M,f)\circ C_\gamma.\] Hence \(f_*\) is smooth. ◻

Proof. The evaluation map \[\varepsilon_c:C(M,N)\times M\to N, \quad (\gamma,p)\mapsto \gamma(p))\] is \(C^{\infty,0}\) with point evaluation \((\varepsilon_c)_p:C(M,N)\to N\), \(\gamma\mapsto \gamma (p)\) smooth for each \(p\in M\). Since the inclusion map \(J:\mathcal{F}(M,N)\to C(M,N)\) is smooth, we have \(\varepsilon=\varepsilon_c\circ (J,\text{Id}_M)\) and \(\varepsilon_p=(\varepsilon_c)_p \circ J\) for each \(p\in M\). ◻

Proof. If \(W\in \mathcal{U}\) is relatively compact and \(z\in \mathbb{R}^n\), consider the constant function \[c_z\colon W\to\mathbb{R}^n,\quad x\mapsto z.\] Then \(c_z\in \mathcal{F}(W,\mathbb{R}^n)\). In fact, Definition 5 c) provides \(V\in \mathcal{U}\) such that \(\overline{W}\subseteq V\). Then \(\eta\colon V\to \mathbb{R}^n\), \(x\mapsto 0\) is in \(\mathcal{F}(V,\mathbb{R}^n)\). The map \(f\colon V\times\mathbb{R}^n\to\mathbb{R}^n\), \((x,y)\mapsto z\) is smooth, whence \(c_z=f\circ (\text{id}_W,\eta|_W)\in \mathcal{F}(W,\mathbb{R}^n)\) by the pushforward axiom.
For each \(z\in N\), the constant function \[\zeta_z\colon M\to N,\quad p\mapsto z\] is in \(\mathcal{F}(M,N)\). In fact, if \(p\in M\), \(\phi_M\colon U_M\to V_M\) is chart for \(M\) around \(p\) and \(\phi_N\colon U_N\to V_N\) a chart for \(N\) around \(\zeta_z(p)=z\), then Definition 5 c) provides a relatively compact \(\phi_M(p)\)-neighborhood \(W\subseteq V_M\) such that \(W\in \mathcal{U}\). After replacing \(\phi_M\) with its restriction to a map \(\phi_M^{-1}(W)\to W\), we may assume that \(V_M\in \mathcal{U}\) and \(V_M\) is relatively compact. Now \(\phi_N\circ \zeta_z\circ \phi_M^{-1}\) is the constant function \(W\to\mathbb{R}^n\), \(x\mapsto \phi_N(z)\), which is in \(\mathcal{F}(W,\mathbb{R}^n)\) as observed above. Thus \(\zeta_z\in \mathcal{F}(M,N)\). In particular, for each \(y\in N\) and \(z\in T_yN\), the constant function \[C_z\colon M\to T_yN,\quad v\mapsto z\] is an element of \(\mathcal{F}(M,T_yN)\). Since \(T_yN\) is a finite-dimensional vector space, the linear map \[\label{returnto} C\colon T_yN\to \mathcal{F}(M,T_yN),\quad z\mapsto C_z\tag{4}\] is continuous. Given \(y\in N\), consider the constant function \(\zeta_y\colon M\to N\), \(p\mapsto y\), we define the vector space \[\Gamma_\mathcal{F}(\zeta_y):= \{\tau\in \mathcal{F}(M,TN): (\forall p\in M)~ \tau(p)\in T_{\zeta_y(p)}N=T_yN \}.\] We show that \[\mathcal{F}(M,T_yN)\subseteq \Gamma_\mathcal{F}(\zeta_y)\] with continuous linear inclusion map. The inclusion map \(\iota\colon T_yN\to TN\) being smooth, for each \(\tau\in \mathcal{F}(M,T_yN)\) we get \[\tau=\iota\circ \tau =\mathcal{F}(M,\iota)(\tau)\in \mathcal{F}(M,TN)\] by Lemma 12. Moreover, \(\mathcal{F}(M,\iota)\) (and hence also its co-restriction \(j\) to \(\Gamma_\mathcal{F}(\zeta_y)\)) is continuous, by Proposition 29.
Let \(\Sigma\colon \Omega\to N\) be a local addition for \(N\) and notation as in Definition 24 and Remark 26. We have \(V\subseteq \Omega\) for an open \(0\)-neighborhood \(V\subseteq T_yN\). Then \(U_N:=\Sigma(V)\) is an open \(y\)-neighborhood in \(N\) and \(\psi:=\Sigma|_V^{U_N}\colon V\to U_N\) is a \(C^\infty\)-diffeomorphism with \[\psi^{-1}(u)=\theta_N^{-1}(y,u)\] for \(u\in U_N\). If \(\alpha\colon T_yN\to \mathbb{R}^n\) is an isomorphism of vector spaces, then \(V_N:=\alpha(V)\) is open in \(\mathbb{R}^n\) and \(\phi_N(u):=\alpha(\psi^{-1}(u))\) defines a chart \(\phi_N\colon U_N\to V_N\) of \(N\). For each \(v\in V_N\), we have for each \(q\in M\) \[(\zeta_y(q),\zeta_{\phi_N^{-1}(v)}(q))=(y,\phi_N^{-1}(v))=(y,\psi(\alpha^{-1}(v))) \in \{y\}\times U_N\subseteq \Omega'\] with \[\theta^{-1}_N(y,\psi(\alpha^{-1}(v)))=\psi^{-1}(\psi(\alpha^{-1}(v)))=\alpha^{-1}(v).\] Thus \(\zeta_{\phi_N^{-1}(v)}\in \mathcal{U}_{\zeta_y}\) and \[\Psi_{\zeta_y}^{-1}(\zeta_{\phi_N^{-1}(v)})=\theta_N^{-1}\circ \left(\zeta_y,\zeta_{\phi_N^{-1}(v)}\right)\] is the constant function \(C_{\alpha^{-1}(v)}\). Hence \[\Psi_{\zeta_y}^{-1}\circ \zeta \circ \phi_N^{-1}=j\circ C\circ \alpha^{-1}|_{V_N},\] which is a smooth function. Thus \(\zeta\) is smooth.
Fix \(p\in M\). The point evaluation \(\varepsilon_p\colon \mathcal{F}(M,N)\to N\), \(\gamma\mapsto \gamma(p)\) is smooth and hence continuous. Since \(\varepsilon_p\circ \zeta=\text{id}_N\), we deduce that \((\zeta|^{\zeta(N)})^{-1}=\varepsilon_p|_{\zeta(N)}\) is continuous. Thus \(\zeta\) is a homeomorphism onto its image. ◻

Proof. Let \(\Sigma:\Omega \to N\) be a normalized local addition of \(N\) in sense of [3]. Since \(\Gamma_{\mathcal{F}}(\gamma)\) is a vector space, we identify its tangent bundle with \(\Gamma_{\mathcal{F}}(\gamma)\times \Gamma_{\mathcal{F}}(\gamma)\). Let \(\Psi_\gamma :\mathcal{V}_\gamma \to \mathcal{U}_\gamma\) be a chart around \(\gamma\) such that \(\Psi_\gamma (0) = \gamma\), then \[T\Psi_\gamma : T\mathcal{V}_\gamma \simeq \mathcal{V}_\gamma\times \Gamma_{\mathcal{F}}(\gamma) \to T \mathcal{F}(M,N)\] is a diffeomorphism onto its image. Moreover, \[T_0\Psi_\gamma : \{0\}\times \Gamma_{\mathcal{F}}(\gamma) \to T_\gamma \mathcal{F}(M,N)\] is an isomorphism of topological vector spaces. We will show that \[\Theta_\gamma \circ T\Psi_\gamma (0,\sigma) = \sigma\] for each \(\sigma\in \Gamma_{\mathcal{F}}(\gamma)\). Which is equivalent to show that \[T\varepsilon_p \circ T\Psi_\gamma (0,\sigma) = \sigma(p) \quad \text{ for all }p\in M.\] Working with the geometric point of view of tangent vectors, we see that \((0,\sigma)\) is equivalent to the curve \([s\mapsto s\sigma]\). Hence, for each \(p\in M\) we have \[\begin{align} T\varepsilon_p \circ T\Psi_\gamma (0,\sigma) &= T\varepsilon_p \circ T\Psi_\gamma ([s\mapsto s\sigma ]) \\ &= T\varepsilon_p ([s\mapsto \Psi_\gamma(s\sigma)]) \\ &= T\varepsilon_p ([s\mapsto \Sigma (s\sigma)])\\ &=[s\mapsto \Sigma|_{T_{\gamma(p)}N} (s\sigma(t))] \\ &= T_0\Sigma|_{T_{\gamma(p)}N} ([s\mapsto s\sigma(t)]). \end{align}\] Since \(\Sigma\) is normalized we have \(T_0\Sigma|_{T_{\gamma(p)}N}=\text{id}_{T_{\gamma(p)}N}\) and \[T\varepsilon_p \circ T\Psi_\gamma (0,\sigma) = \sigma (p).\] In consequence, for each \(\sigma\in \Gamma_{\mathcal{F}}(\gamma)\), there exists a \(v\in T_\gamma \mathcal{F}(M,N)\) with \(v=T\Psi_\gamma (0,\sigma)\) such that \[\Theta_\gamma (v) = \sigma.\] Moreover, the function \[\Theta_\gamma(v):M\to TN,\quad p\mapsto \Theta_N(v)(p) = \sigma(p)\in T_{\gamma(p)}N\] is in \(\mathcal{F}(M,TN)\) with \(\pi_{N}\circ \Theta_\gamma(v) =\gamma\), making the map \(\Theta_\gamma\) an isomorphism of topological vector spaces. ◻

Proof. Let \(\Sigma_1:\Omega_1\to N_1\) be a local addition on \(N_1\) and \(\gamma \in \mathcal{F}(M,N_1)\).
If \(\Psi_\gamma:\mathcal{V}_\gamma\to \mathcal{U}_\gamma\) is a chart on \(\gamma\) such that \(\Psi_\gamma (0)= \gamma\), we consider the isomorphism of vector space \[T\Psi_\gamma : \{0\}\times \Gamma_{\mathcal{F}}(\gamma) \to T_\gamma \mathcal{F}(M,N_1).\] For \(p\in M\) we denote the point evaluation in \(\varepsilon_p^i:\mathcal{F}(M,N_i)\to N_i\) for \(i\in \{1,2\}\), then for each \(\sigma \in \Gamma_{\mathcal{F}}(\gamma)\) we have \[\begin{align} T\varepsilon_p^2\circ T\mathcal{F}(M, f) \circ T\Psi_\gamma (0,\sigma) &= T\varepsilon_p^2\circ T\mathcal{F}(M,f) \circ T\Psi_\gamma([s\mapsto s\sigma]) \\ &= T\varepsilon_p^2\circ T\mathcal{F}(M,f) ([s\mapsto \Sigma_1 \circ s\sigma])\\ &= T\varepsilon_p^2 ([s\mapsto f \circ \Sigma_1 \circ s\sigma])\\ &= [s\mapsto \varepsilon_p^2\left(f \circ \Sigma_1 \circ s\sigma\right)] \\ &= [s\mapsto f \circ \Sigma_1 (s\sigma(p))]\\ &= Tf \circ T_0\Sigma_1|_{T_{\gamma(p)}N_1} ([s\sigma(p)]) \\ &= Tf([s\mapsto s\sigma(p)])\\ &= Tf(\sigma(p)) \\ &= \mathcal{F}(M,Tf)\circ T\varepsilon_p^1 \circ T\Psi_\gamma (0,\sigma). \end{align}\] Hence \[\Theta_{N_2}\circ T\mathcal{F}(M,f) =\mathcal{F}(M,Tf)\circ \Theta_{N_1}.\] ◻

Proof. For the open sets \(\Omega\subseteq TN\) and \(\Omega':=(\pi_N,\Sigma)(\Omega)\subseteq N\times N\) we define the open sets \[\mathcal{F}(M,\Omega):=\{\sigma\in \mathcal{F}(M,TN) : \sigma(M)\subseteq \Omega\}\] and \[\mathcal{F}(M,\Omega') := \{ \alpha\in \mathcal{F}(M,N\times N) : \alpha(M)\subseteq \Omega' \}.\] Let \(\gamma\in \mathcal{F}(M,N)\), we define \(\sigma_\gamma:M\to TN, p\mapsto 0_{\gamma(p)}\). Then \(\sigma_\gamma\in \Gamma_\mathcal{F}(\gamma)\) and the zero-section is given by \[\mathcal{F}(M,N)\to \mathcal{F}(M,TN),\quad \sigma\mapsto \sigma_\gamma.\] Moreover, we see that \[\mathcal{F}(M,\Sigma)(\sigma_\gamma)(p)=\left(\Sigma\circ \sigma_\gamma\right)(p)=\Sigma(0_{\gamma(p)})=\gamma(p)\] hence \(\mathcal{F}(M,\Sigma)(\sigma_\gamma)=\gamma\) for each \(\gamma\in \mathcal{F}(M,N)\).
Since \((\pi_N,\Sigma):\Omega\to \Omega'\) is a \(C^\infty\)-diffeomorphism, by Proposition 29, we can define the \(C^\infty\)-diffeomorphism \[\Theta:=\mathcal{F}\left(M,(\pi_N,\Sigma)\right): \mathcal{F}(M,\Omega)\to \mathcal{F}(M,\Omega'), \quad \sigma\mapsto (\pi_{N},\Sigma)\circ \sigma\] with inverse given by \[\Theta^{-1}:=\mathcal{F}\left(M,(\pi_N,\Sigma)^{-1}\right): \mathcal{F}(M,\Omega')\to \mathcal{F}(M,\Omega), \quad \alpha\mapsto (\pi_{N},\Sigma)^{-1}\circ \alpha\] Hence \(\mathcal{F}(M,\Sigma)\) is a local addition on \(\mathcal{F}(M,N)\). ◻

Proof. This is direct consequence of the properties of the supremum. ◻

Proof. By relative compactness of \(W\), we can consider a finite open cover of convex subsets \((W_i)_{i=1}^k\) for \(\overline{W}\) such that \(\Theta|_{W_i}\) is Hölder continuous and \(\eta\circ \Theta|_{W_i} \in \mathcal{F}_\lambda(W_i,\mathbb{R})\) for each \(i\in\{1,...,k\}\) and \(\eta\in \mathcal{F}_\lambda (V,\mathbb{R})\). Therefore \(\gamma\circ \Theta|_W\in \mathcal{F}(W,E)\). ◻

Proof. Let \(\eta \in \mathcal{F}_\lambda(U,\mathbb{R})\). Since the function \(h\) is smooth with compact support, is \(\lambda\)-Hölder continuous and the product \(h\eta\) is in \(\mathcal{F}_\lambda(U,\mathbb{R})\). ◻

Proof. Let \(\Delta_V\) denote the diagonal set of \(V\times V\). For each \(\tau\in \mathcal{F}_\lambda(V,\mathbb{R})\), we define the function \[h_\tau:(V\times V)\setminus \Delta_V \to \mathbb{R},\quad (x,y)\mapsto h_\tau(x,y):=\frac{\tau(x)-\tau(y)}{\lVert x-y\lVert^{\lambda}}.\] Then \(h_\tau \in BC((V\times V)\setminus \Delta_V,\mathbb{R})\) with \(\lVert h_\tau\lVert_\infty = \lVert \tau \lVert_\lambda\), hence the linear map \[\mathcal{F}_\lambda(V,\mathbb{R})\to BC((V\times V)\setminus \Delta_V,\mathbb{R}), \quad \tau\mapsto h_\tau\] is continuous linear. Let us consider the map \[H:\mathcal{F}_\lambda(V,\mathbb{R})\to BC((V\times V)\setminus \Delta_V,\mathbb{R}), \quad \tau\mapsto h_\tau\] then \(H\) is continuous. This enable us to define the linear map \[\Phi:\mathcal{F}_\lambda(V,\mathbb{R})\to BC(V,\mathbb{R})\times BC((V\times V)\setminus \Delta_V,\mathbb{R}),\quad \tau\mapsto (\tau,H(\tau))\] which is a topological embedding with closed image. Therefore, if the map \(\tilde{f}\) makes sense, its continuity is equivalent to the continuity of \[F: \mathcal{F}_\lambda(V,\mathbb{R}^\ell) \to BC(V,\mathbb{R})\times BC((V\times V)\setminus \Delta_V,\mathbb{R}),\quad \eta \mapsto \left(f\circ \eta, H(f\circ\eta)\right).\] First we will show that makes sense, i.e., \(F(\eta)\in BC(V,\mathbb{R})\times BC((V\times V)\setminus \Delta_V,\mathbb{R})\) for each \({\eta \in \mathcal{F}_\lambda(V,\mathbb{R}^\ell)}\). Since the inclusion map \(J:\mathcal{F}_\lambda(V,\mathbb{R}^\ell)\to BC(V,\mathbb{R}^\ell)\) and the map \[BC(V,\mathbb{R}^\ell)\to BC(V,\mathbb{R}),\quad \eta\mapsto f\circ \eta\] are continuous, the first component of \(F\) \[F_1:\mathcal{F}_\lambda(V,\mathbb{R}^\ell)\to BC(V,\mathbb{R}),\quad \eta \mapsto f\circ \eta\] is continuous. Let us consider the second component of \(F\) \[F_2: \mathcal{F}_\lambda(V,\mathbb{R}^\ell) \to BC((V\times V)\setminus \Delta_V,\mathbb{R}),\quad \eta \mapsto H(f\circ\eta).\] Let \(\eta \in \mathcal{F}_\lambda(V,\mathbb{R}^\ell)\), then \(F_2(\eta)\) is clearly continuous. We will show that \(F_2(\eta)\) is bounded. For \((x,y)\in V\times V \setminus \Delta_V\) we have \[F_2(\eta)(x,y) = H(f\circ \eta)(x,y) = \frac{f(\eta(x))-f(\eta(y))}{\lVert x-y\lVert^\lambda}.\] Since \(V\) is relatively compact, the set \(\eta(V)\) can be contained on an open ball \(B_{R_\eta}(0)\) for a constant \(R_\eta>0\) large enough. By smoothness, the map \(f\) verifies \[\lvert f(u)-f(v)\lvert\leq L_{f,\eta} \lVert u-v \lVert,\quad u,v\in \overline{B_{R_\eta}(0)},\] for some constant \(L_{f,\eta}>0\). Therefore \[\lVert F_2(\eta)\lVert_\infty \leq L_{f,\eta} \lVert \eta \lVert_\lambda.\] Then \(F_2(\eta)\in BC((V\times V)\setminus \Delta_V,\mathbb{R})\). Now we will show that \(F_2\) is continuous in \(\eta\in F_\lambda(V,\mathbb{R}^\ell)\). Let \(\delta>0\) and \(\gamma\in F_\lambda(V,\mathbb{R}^\ell)\) such that \[\lVert \eta - \gamma\lVert_{\mathcal{F}_\lambda} := \lVert \eta-\gamma\lVert_\infty + \lVert \eta-\gamma\lVert_\lambda\leq \delta.\] Then for each \(z\in V\) we have \[\lVert \eta(z)-\gamma(z) \lVert \leq \delta,\] which mean that \(\gamma(z)\in B_\delta(\eta(z))\). Therefore \[\gamma(V)\subseteq \bigcup_{z\in V} B_\delta (\eta(z)).\] Let \(R_\eta>0\) the constant which verifies \(\eta(V)\subseteq B_{R_\eta}(0)\), then \(B_\delta(\eta(z))\subseteq B_{R_\eta+\delta}(0)\) for each \(z\in V\). In consequence, \(\gamma(V)\) and \(\eta(V)\) are contained in \(B_{R_\eta+\delta}(0)\) and by smoothness of \(f\), there exists a constant \(G_{f,\eta}>0\) such that \[\lvert df(u_1,v_1)-df(u_2,v_2)\lvert\leq G_{f,\eta} \lVert (u_1,v_1)- (u_2,v_2) \lVert=G_{f,\eta}(\lVert u_1-u_2\lVert+\lVert v_1-v_2\lVert),\] for each \((u_1,v_1), (u_2,v_2)\in \overline{B_{R_\eta+\delta}(0)}\times \overline{B_{R_\eta+\delta}(0)}\). By the mean value theorem, we have \[f(u_1)-f(u_2)=\int_0^1 df(u_2+t(u_1-u_2),u_1-u_2)dt,\quad u_1, u_2\in \overline{B_{R_\eta+\delta}}.\] Hence, if \(\omega:= \lvert F_2(\eta)(x,y)-F_2(\gamma)(x,y) \lvert\) then \[\begin{align} \omega&= \left\lvert \frac{f(\eta(x))-f(\eta(y))}{\lVert x-y\lVert^\lambda}-\frac{f(\gamma(x))-f(\gamma(y))}{\lVert x-y\lVert^\lambda}\right\lvert \\ &= \left\lvert\int_0^1\left( df\left(\eta(y)+t(\eta(x)-\eta(y)),\frac{\eta(x)-\eta(y)}{\lVert x -y\lVert^\lambda}\right)- df\left(\gamma(y)+t(\gamma(x)-\gamma(y)),\frac{\gamma(x)-\gamma(y)}{\lVert x -y\lVert^\lambda}\right)\right)dt\right\lvert\\ &\leq G_{f,\eta} \int_0^1 \left\lVert \left(\eta(y)+t(\eta(x)-\eta(y)),\frac{\eta(x)-\eta(y)}{\lVert x -y\lVert^\lambda}\right)-\left(\gamma(y)+t(\gamma(x)-\gamma(y)),\frac{\gamma(x)-\gamma(y)}{\lVert x -y\lVert^\lambda}\right)\right\lVert dt\\ &\leq G_{f,\eta}\left( \int_0^1 \lVert t(\eta(x)-\gamma(x))+(1-t)(\eta(y)-\gamma(y))\lVert dt + \frac{\lVert\big(\eta(x)-\gamma(x)\big)-\big(\eta(y)-\gamma(y)\big)\lVert}{\lVert x-y\lVert^\lambda} \right) \\ &\leq G_{f,\eta}( \lVert \eta-\gamma \lVert_\infty + \lVert \eta-\gamma \lVert_\lambda ) \\ & \leq G_{f,\eta}\delta. \end{align}\] If \(\varepsilon=G_{f,\eta}\delta\), we have \[\lVert F_2(\eta)-F_2(\gamma)\lVert_\infty \leq \varepsilon.\] Therefore, the map \(F_2\) is continuous and in consequence, the map \(\tilde{f}\) is continuous. ◻

Proof. First let assume that \(U=\mathbb{R}^m\). Let \(\text{id}:V\to \mathbb{R}^m\) be the identity map, then \(\text{id}\in \mathcal{F}_\lambda(V,\mathbb{R}^m)\) and by Lemma 17, the map \[F_\lambda (\mathbb{R}^m,\mathbb{R}^n)\to F_\lambda (V,\mathbb{R}^m\times \mathbb{R}^n),\quad \eta\mapsto (\text{id},\eta|_V)\] is continuous. If \(\ell=m+n\), by Lemma 20, the map \[F_\lambda (V, \mathbb{R}^m\times \mathbb{R}^n)\to F_\lambda (V,\mathbb{R}),\quad \beta \mapsto f\circ \beta\] is continuous. Therefore \(f_*\) is just the composition of continuous mappings.
Let assume that \(U\neq \mathbb{R}^m\). Let \(\chi:\mathbb{R}^m\to\mathbb{R}\) be a cut-off function for \(\overline{V}\) supported in \(U\) (see e.g. [6]); we define \[g:\mathbb{R}^m\times \mathbb{R}^n\to \mathbb{R},\quad (x,y) \mapsto \left\{\begin{array}{ll} \chi(x)f(x,y),& \text{if } x\in U \\ 0, & \text{if } x\in \mathbb{R}^m\setminus\text{supp}(\chi) \end{array}\right.\] Then \(g\) is smooth and, as before, the map \[g_*:\mathcal{F}_\lambda (\mathbb{R}^m\times \mathbb{R}^n,\mathbb{R}^n)\to \mathcal{F}_\lambda (V,\mathbb{R}),\quad \eta\mapsto g_*(\eta)=g\circ (\text{id},\eta|_V)\] is continuous. Moreover, for each \(\eta \in \mathcal{F}_\lambda(U,\mathbb{R}^n)\) and \(x\in V\) we have \[\begin{align} g_*(\eta)(x)&=g\circ(id,\eta|_V)(x) \\ &= g(x,\eta|_V(x)) \\ &= \chi(x)f(x,\eta|_V (x)) \\ &= f(x,\eta|_V(x)) \\ &= f_*(\eta)(x), \end{align}\] whence \(g_*=f_*\). ◻

Proof. Let \(\gamma\in C^{0,\lambda}(M,N)\), then for each \(p\in M\), there exists the charts \(\phi_M:U_M\to V_M\) of \(M\) and \(\phi_N:U_N\to V_N\) of \(N\), such that \(p\in U_M\), \(\gamma(U_M)\subseteq U_N\) and \(\phi_N\circ \gamma\circ \phi_M^{-1}\in \mathcal{F}_\lambda(V_M,\mathbb{R}^n)\). For each \(U\in \mathcal{U}\), it is known that for \(\beta\leq \lambda\) the linear operator \[I_{U}: \mathcal{F}_\lambda(U,\mathbb{R}^n)\to \mathcal{F}_\beta(U,\mathbb{R}^n),\quad \tau \mapsto \tau\] is continuous. In particular, we have \[I_{V_M}(\phi_N\circ \gamma\circ \phi_M^{-1})=\phi_N\circ \gamma\circ \phi_M^{-1}\in \mathcal{F}_\beta(V_M,\mathbb{R}^n).\] Therefore \(\gamma\in C^{0,\beta}(M,N)\). Now, we consider the charts \((\mathcal{U}_\gamma, \Psi_\gamma^{-1})\) and \((\mathcal{U}_{\iota(\gamma)}, \Psi_{\iota(\gamma)}^{-1})\) in \(\gamma \in C^{0,\lambda}(M,N)\) and \(\iota(\gamma) \in C^{0,\beta}(M,N)\) respectively, then the map \[\Psi_{\iota(\gamma)}^{-1} \circ\iota \circ \Psi_\gamma: \Psi_\gamma^{-1}\left( \mathcal{U}_\gamma \cap \iota^{-1}(\mathcal{U}_{\iota(\gamma)})\right)\to \Psi_{\iota(\gamma)}\left( \mathcal{U}_\gamma \cap \iota^{-1}(\mathcal{U}_{\iota(\gamma)})\right)\] given by \[\Psi_{\iota(\gamma)}^{-1} \circ\iota \circ \Psi_\gamma (\sigma) = (\pi_{N},\Sigma_N)^{-1} \circ \Big(\iota(\gamma) , \iota (\Sigma_N \circ \sigma) \Big)\] is just a restriction of the map \[\tilde{\iota}:\Gamma_{\mathcal{F}_\lambda}(\eta)\to \Gamma_{\mathcal{F}_\beta}(\iota(\eta)),\quad \sigma\mapsto \sigma,\] which is continuous by Proposition 15 and continuity of the maps \(\{I_U\}_{U\in\mathcal{U}}\). ◻