Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of smooth sections of $E$ is naturally a Fréchet space (with the seminorms given in terms of multiple derivatives of sections).

Now suppose that all the objects above are real analytic and let $\Gamma^\omega(M,E)$ be the space of real analytic sections of $E$.

My question is: is $\Gamma^\omega(M,E)$ a Fréchet space in some “natural” way? If not, what is the “richest” structure one can put in this vector space? For instance, I would like to know if it is a Baire and metrizable topological vector space.