Let $X$ be a set equipped with the discrete topology and consider the space $C\_c(X)$ of complex valued functions of finite support. It comes with a natural locally convex inductive limit topology. Is this space reflexive or not?



Is $C_c(X)$ reflexive?
Yes, it is. Indeed,


1º) As inductive limit over finite sets $K\subset X$ of the finite-dimensional vector spaces $C\_c(K)$, on which *every* seminorm is continuous, $C\_c(X)$ has the topology defined by *all* seminorms on it, a.k.a. the *finest locally convex topology* (see e.g. [Robertson & Robertson 1964](http://www.ams.org/mathscinet-getitem?mr=162118), [p. 22](https://books.google.com/books?id=Ea48AAAAIAAJ&pg=PA22)).


2º) Any vector space, equipped with said finest locally convex topology, is Montel (ibid., [p. 75](https://books.google.com/books?id=Ea48AAAAIAAJ&pg=PA75)) and hence reflexive (ibid., [p. 74](https://books.google.com/books?id=Ea48AAAAIAAJ&pg=PA74)).


