Definition 28.3.2 (Holomorphic on $\complex_{\infty}$).label Let $E$ be a separated locally convex space and $f \in C(\complex_{\infty}; E)$, then $f$ is holomorphic at $\infty$ if $z \mapsto f(z^{-1})$ (under the identification that $1/0 = \infty$) is holomorphic at $0$.