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 !

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

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