A systematic treatment is proposed for the calculus involving product-type mappings with different degrees of differentiability in the two factors.
The following are the main results of this paper.
{ Theorem A.} Let \(E_1\), \(E_2\) and \(F\) be locally convex spaces, \(U\subseteq E_1\) and \(V\subseteq E_2\) be convex subsets with dense interior, and \(r,s\in \mathbb N_0\cup \{\infty\}\). Then,

(I)

\(\gamma^\vee:U\to C^s(V,F)\),\(x\mapsto \gamma(x,.)\) is \(C^r\) for each \(\gamma\in C^{r,s}(U\times V,F)\), and the map \(\Phi: C^{r,s}(U\times V,F)\to C^r(U,C^s(V,F))\),\(\gamma \mapsto \gamma^\vee\) is linear and a topological embedding;

(II)

if \(U\times V\subseteq E_1\times E_2\) is a \(k\)-space or \(V\) is locally compact, then \(\Phi\) is an isomorphism of topological vector spaces.

{ Theorem B.} Let \(M_1\) and \(M_2\) be smooth manifolds (possibly with rough boundary), modeled on locally convex spaces \(E_1\) and \(E_2\), respectively. Let \(F\) be a locally convex space and \(r,s\in \mathbb N_0\cup \{\infty\}\). Then,

(I)

\(\gamma^\vee\in C^r(M_1,C^s(M_2,F))\), for all \(\gamma\in C^{r,s}(M_1\times M_2,F)\);

(II)

the map \(\Phi: C^{r,s}(M_1\times M_2,F)\to C^r(M_1,C^s(M_2,F))\),\(\gamma\mapsto \gamma^\vee\) is linear and a topological embedding;

(III)

if \(E_1\) and \(E_2\) are metrizable, then \(\Phi\) is an isomorphism of topological vector spaces.

Finally, as an application of this \(C^{r,s}\)-type theory, the parameter dependence of solutions of differential equations in locally convex spaces is discussed.