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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.

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    $\begingroup$ The space or real analytic functions with a suitable locally convex topology has been studied in detail by functional analysts, one reason being that its lc structure doesn't fit into the familiar special classes (Fréchet, Silva,...). Rather than giving explicit references, I would suggest you check out some of the main actors, e.g., André Martineau, Pawel Domański and Dietmar Vogt. $\endgroup$
    – terceira
    Commented Sep 17, 2023 at 5:21
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    $\begingroup$ If $\Omega$ is connected and $K$ is closed ball contained in $\Omega$, then $\|f\|_K=\sup\{|f(x)|:x\in K\}$ is a norm on $C^\omega(\Omega)$. As mentioned in terceira's comment, the correct locally convex topology was introduced by Martineau. $\endgroup$
    – Jochen Wengenroth
    Commented Sep 17, 2023 at 7:17

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