Lineability and algebrability of the set of holomorphic functions with a given domain of existence

Abstract

Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.