# A remark on projective limits of function spaces

## TL;DR

This paper investigates the structure of projective limits of function spaces of holomorphic functions on open subsets of complex Euclidean space, showing they are homeomorphic and linearly isomorphic to the space of all holomorphic functions, regardless of the specific intermediate spaces used.

## Contribution

It demonstrates that the projective limit of certain holomorphic function spaces is always homeomorphic and linearly isomorphic to the space of all holomorphic functions, independent of the chosen intermediate spaces.

## Key findings

- Projective limits of holomorphic function spaces are homeomorphic to H(Omega).
- The isomorphism holds regardless of the intermediate spaces Xm(Omega_m).
- The result applies to various topological spaces of holomorphic functions.

## Abstract

Let Omega subset of C^d be an open set and Km, m = 1, 2, . . . an exhaustion of Omega by compact subsets of Omega. We set Omega_m = int(Km) and let Xm(Omega_m) be a topological space of holomorphic functions on Omega_m between A^ infinity (Omega_m) and H(Omega_m). Then we show that the projective limit of the family Xm(Omega_m), m = 1, 2, . . . , under the restriction maps is homeomorphic and linearly isomorphic to the Frechet space H(Omega), idependently of the choice of the spaces Xm(Omega_m).