Spaceability and algebrability in $(\mathcal {DFC})$-spaces
Volume 160 / 2020
Colloquium Mathematicum 160 (2020), 89-107
MSC: Primary 46G20, 46A04; Secondary 33E99.
DOI: 10.4064/cm7564-1-2019
Published online: 8 January 2020
Abstract
Let $U$ be a connected domain of existence in a $(\mathcal {DFC})$-space and let $\mathcal {E}(U)$ be the set of holomorphic functions on $U$ whose domain of existence is $U$. We construct a $\mathfrak {c}$-dimensional subspace within $\mathcal {E}(U) \cup \{0\}$ and we give sufficient conditions for $\mathcal {E}(U)$ to be strongly algebrable. We also prove that $\mathcal {E}(U) \cup \{0\}$ contains a closed algebra $\mathcal {A}$ as well as a dense algebra $\mathcal {B}$, both containing an infinite algebraically independent set.
