In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function $f: O \to E$, where $O$ is an open subset of the complex plane and $E$ a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.

For $\Omega_1$ a subset of the Riemann sphere, we consider the space of locally holomorphic maps of $\Omega_1$ into $E$ vanishing at infinity if infinity belongs to $\Omega_1$, denoted by $P(\Omega_1,E)$. For two complementary subsets $\Omega_1$ and $\Omega_2$ of the Riemann sphere we prove that given two locally convex topological vector spaces $E$ and $F$ in separating duality, under some general conditions, we can define a separating duality between $P(\Omega_1,E)$ and $P(\Omega_2,F)$.