{title2=$\mathcal O(U)$}

For an <open set> $U\subseteq\mathbb C$, the space $\mathcal O(U)$ consists of all <holomorphic function>[holomorphic functions] on $U$. Its standard topology is locally uniform convergence, equivalently the <compact-open topology> defined by the seminorms $f\mapsto\sup_{z\in K}|f(z)|$ for compact $K\subset U$.

= Analytic function
{synonym}


Space of holomorphic functions

{wiki}

The compact-open topology on a space of functions is generated by uniform control on compact subsets. For scalar-valued functions, its basic seminorms are $f\mapsto\sup_{x\in K}|f(x)|$ as $K$ ranges over compact subsets of the domain.


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 Incoming links: Compact-open topology