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 → 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 Ω_1 a subset of the Riemann sphere, we consider the space of locally holomorphic maps of Ω_1 into E vanishing at infinity if infinity belongs to Ω_1, denoted by P(Ω_1,E). For two complementary subsets Ω_1 and Ω_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(Ω _1,E) and P(Ω_2,F).

- All seminars.
- Seminars for 2018