# Existence of the limit of a bounded analytic function

Let $$X$$ be a Banach space and let $$U\subset\mathbb{C}$$ be open. If we have an analytic function $$f:U\setminus \left\{ 0\right\} \rightarrow X$$ such that $$\underset{x\in U\setminus \left\{ 0\right\} }{\sup }\left\Vert f\left( x\right) \right\Vert <\infty ,$$ can we say that the limit $$\underset{x\rightarrow 0}{\lim }f\left( x\right)$$ exists. Thank you !

• Have you tried adapting a proof from the case where $X=\mathbb C$? – Aweygan Jan 13 at 23:51
• Not yet, but I think that if X is finite dimensional then the limit exists – Djalal Ounadjela Jan 14 at 11:18