Understanding a function space

Question

I was reading a paper on Homogenization theory, where the author uses the spaces of vector valued functions. Let us consider such a space $D[\Omega; C^\infty_P(Y)]$, consisting of all the compactly supported smooth functions $\psi : \Omega \rightarrow C^\infty_P(Y)$. $Y=[0,1]^n$, $\Omega$ is an open set in $R^n$ and $C^\infty_P(Y)$ is the space of all smooth functions on $R^n$ that are Y-periodic. A function $f$ is called Y-periodic if $f(x+ke_i)=f(x)$ for all $x$ in $R^n$ and integers $k$ and for all $i$ $\in$ {1,2,...,n} ; $e_i$'s being standard basis vectors in $R^n$. Note that, the smoothness (differentiablity) is defined on $ \psi$ as Fréchet derivative between the normed spaces $R^n$ and $C^\infty_P(Y)$. The author did not mention which norm to consider on $C^\infty_P(Y)$. Let us take the sup norm on this space. Clearly $\psi(x,.)$ is an element of $C^\infty _P(Y)$ for each $x$ in $\Omega$. Also, $\psi (x,y) $ is a real number for each $x$ in $ \Omega$ and $y$ in $R^n$. So, we can also think of such a $\psi$ as a real valued function defined on $\Omega \times R^n$. It seems that the author is treating such a $\psi(x,y)$ as a real valued SMOOTH function defined on $ \Omega \times R^n$. But how to argue that $\psi(x,y)$ is smooth on $ \Omega \times R^n$? Thank you.