This is a comment but will be too long. You start with the following setup: a pair of Fréchet spaces (the holomorphic function on the complex plane, resp. on the complement of the real line there) and a natural continuous, linear injection of the former into the latter. The object of interest is then the corresponding quotient space. This is precisely the setting of the theory of quotient spaces (in your case quotient Fréchet spaces) which was introduced and studied by Lucien Waelbroeck and associates in the second half of the last century, under the name of $q$-spaces (for quotient). The setting is a suitable category from functional analysis (Banach spaces, Fréchet spaces, complete convex bornological spaces, $\dots$). One then defines a new category whose objects are pairs $E,E_0$ and a continuous injective morphism from $E_0$ into $E$. The morphisms are defined in the natural way. The objects of interest are the so-called underlying spaces $E/E_0$.

Waelbroeck began his investigations in the 60‘s but the first publication that I know of dates from 1977 („Quotient Banach spaces“, Warsaw). He considered the case of cbsˋs in the monograph „Bornological quotients“ (1998, with G. Noël). Both are easily available online.

The motivation came from operator theory—spectral theory and functional calculus.

This is a comment but will be too long. You start with the following setup: a pair of Fréchet spaces (the holomorphic functions on the complex plane, resp. on the complement of the real line there) and a natural continuous, linear injection of the former into the latter. The object of interest is then the corresponding quotient space. This is precisely the setting of the theory of quotient spaces (in your case quotient Fréchet spaces) which was introduced and studied by Lucien Waelbroeck and associates in the second half of the last century, under the name of $q$-spaces (for quotient). The setting is a suitable category from functional analysis (Banach spaces, Fréchet spaces, complete convex bornological spaces, $\dots$). One then defines a new category whose objects are pairs $E,E_0$ and a continuous injective morphism from $E_0$ into $E$. The morphisms are defined in the natural way. The objects of interest are the so-called underlying spaces $E/E_0$.

Waelbroeck began his investigations in the 60‘s but the first publication that I know of dates from 1977 („Quotient Banach spaces“, Warsaw). He considered the case of cbsˋs in the monograph „Bornological quotients“ (1998, with G. Noël). Both are easily available online.

The motivation came from operator theory—spectral theory and functional calculus.