Hello, this question may be simple but I couldn't find a reference.

Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of bounded real analytic functions from $\Omega$ to $F$. Now I would like to know if there is a natural way to define a metric on $C_b^\omega(\Omega,F)$, that makes the space complete.

Concretely, I have a series of real analytic functions that converge uniformly as well as their Frechet derivatives and now I would like to know if their limit is analytic again.

My first idea was to show that $C_b^\omega(\Omega,F)$ is a closed subspace of $C_b^\infty(\Omega,F)$. Here $C_{b}^{\infty}(\Omega,F)$ is the set of infinitely Frechet-differentiable functions equipped with the usual set of seminorms (i.e. $\Vert f\Vert_k:=\Vert D^k f\Vert_\infty$) defining a Frechet space and thus a metric $d(f,g):=\sum_{k=1}^\infty 2^{-k}\frac{\Vert f-g\Vert_k}{1+\Vert f-g\Vert_k}$ which makes $C_{b}^{\infty}(\Omega,F)$ a complete metric space.

Actually, I have no idea if this is true at all. Could someone please confirm if this is the right thing to prove or not?

Regards, Mirko

unionof a countable family of Banach spaces, not a projective limit. I'm pretty sure that it isn't complete either (since an arbitrary analytic function should be a limit of bounded ones, but I'd need to think a bit harder about the topologies involved to be sure). $\endgroup$