The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet space.
I am wondering if there is a way to define a norm on the space of real analytic real-valued functions defined on $\mathbb{R}^n$, even if not in a canonical way.
I have no patient to write out all details, so I will just give something based on intuition
Define distance (not norm) on power series space $\text{dist}(A,B) = \sup |A_n - B_n|^{\frac{1}{n}}$ so that $\text{dist}(A(x),B(x)) < \epsilon \implies |\sum_{n \ge 1} A_n (v^n) - \sum_{n \ge 1} B_n (v^n)| \le \sum_{n \ge 1} |A_n - B_n| |v|^n = \frac{1}{1 - \epsilon |v|} - 1 \to 0 \text{ as } \epsilon \to 0$
Remark (off topic): This inspired us to define analytic type Sobolev space, i.e. define the integral distance $\sup_n \int |\frac{1}{n!} d^n f - \frac{1}{n!} d^n g|^{1/n}$
Change the base point of power series $A_m (x + \Delta) = \sum_{n \ge m} \binom{n}{m} A_n \Delta^{n-m}$
Compare at point $x + \Delta$
$$|A_m (x + \Delta) - B_m (x + \Delta)| \le \sum_{n \ge m} \binom{n}{m, n-m} |A_n - B_n| |\Delta|^{n-m}$$
Use $|A_n - B_n| < \epsilon^n$ and $n = p+m$ and definition of Taylor series
$$|A_m (x + \Delta) - B_m (x + \Delta)| \le \sum_{p \ge 0} \binom{p+m}{m,p} \epsilon^m |\epsilon \Delta|^p = \epsilon^m \frac{1}{(1 - |\epsilon \Delta|)^{m+1}}$$
$$|A_m (x + \Delta) - B_m (x + \Delta)|^{\frac{1}{m}} \le \epsilon \frac{1}{(1 - |\epsilon \Delta|)^{1 + \frac{1}{m}}} \le \frac{1}{(1 - |\epsilon \Delta|)^{2}}$$
Let $r < \min{R_A (x), R_B (x)}$ where $R(x)$ is radius of convergence at $x$, we have control
$$\lim_{\text{dist}(A(x),B(x)) \to 0} \sup_{|\Delta| \le r} |\sum |A_n (x + \Delta) - B_n (x + \Delta)| |v|^n| = 0$$
After finite step $x_i = x + \Delta_1 + \cdots + \Delta_i$, in compact set $\cup_{i = 1 .. N} \overline{\mathbb{B}} (x_i, r_i)$, it can still be control by the distance $\text{dist}(A(x),B(x))$ at $x$
Let $f : D_f \to \mathbb{R}$ be analytic, where $D_f$ is maximal continuation
Approximate $f$ in by
forall $\epsilon$, forall compact $D \subset D_f$, exist analytic $g : D_g \to \mathbb{R}$ ($D_g$ is maximal continuation) with $D \subset D_g$, so that $\sup_{x \in D} \text{dist} (\frac{1}{n!} d^n f (x) , \frac{1}{n!} d^n g (x)) < \epsilon$
This will define a net for space of analytic function
This is also similar to the definition of compact-open topology of space of continuous function
