The Fréchet space of holomorphic functions on the unit disc

If X is a topological space and p ∈ X, a local basis at p is a set B of open neighborhoods of p such that if U is an open neighborhood of p then there is some U0 ∈ B that is contained in U . We emphasize that to say that a topological vector space (X, τ) is normable is to say not just that there is a norm on the vector space X, but moreover that the topology τ is induced by the norm. A topological vector space over C is a vector space X over C that is a topological space such that singletons are closed sets and such that vector addition X × X → X and scalar multiplication C × X → X are continuous. It is not true that a topological space in which singletons are closed need be Hausdorff, but one can prove that every topological vector space is a Hausdorff space. For any a ∈ X, we check that the map x 7→ a+x is a homeomorphism. Therefore, a subset U of X is open if and only if a+U is open for all a ∈ X. It follows that if X is a vector space and B is a set of subsets of X each of which contains 0, then there is at most one topology for X such that X is a topological vector space for which B is a local basis at 0. In other words, the topology of a topological vector space is determined by specifying a local basis at 0. A topological vector space X is said to be locally convex if there is a local basis at 0 whose elements are convex sets.