\section{Spaces of Holomorphic Functions}
\label{section:holomorphic-function-space}

\begin{definition}[Space of Holomorphic Functions]
\label{definition:holomorphic-function-space}
    Let $E$ be a complete separated locally convex space over $\complex$ and $U \subset \complex$ be open, then $H(U; E)$ is the \textbf{space of $E$-valued holomorphic functions on $U$}, equipped with the topology of uniform convergence on compact sets. 
\end{definition}

\begin{proposition}
\label{proposition:holomorphic-complete}
    Let $E$ be a complete separated locally convex space over $\complex$ and $U \subset \complex$ be open, then $H(U; E)$ is complete. 
\end{proposition}
\begin{proof}
    By \hyperref[Cauchy's estimate]{corollary:cauchy-estimate}, uniform convergence on compact sets is equivalent to uniform convergence of derivatives of all orders on compact sets. Since $U$ is locally compact, uniform limits of holomorphic functions are holomorphic by \autoref{theorem:differentiable-uniform-limit}.
\end{proof}

\begin{theorem}[Montel]
\label{theorem:montel}
    Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, and $\cf \subset H(U; E)$, then the following are equivalent:
    \begin{enumerate}[label=(B\arabic*)]
        \item $\cf$ is equicontinuous, and for each $x \in U$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is bounded.
        \item $\cf$ is bounded in $H(U; E)$.
    \end{enumerate}

    and the following are equivalent:
    \begin{enumerate}[label=(C\arabic*)]
        \item $\cf$ is relatively compact in $H(U; E)$. 
        \item $\cf$ is bounded in $H(U; E)$, and for each $x \in U$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is relatively compact.  
    \end{enumerate}
\end{theorem}
\begin{proof}
    (B1) $\Rightarrow$ (B2): Let $K \subset U$ be compact and $V \in \cn_E(0)$ be circled. Since $\cf$ is equicontinuous, for each $x \in K$, there exists $U_x \in \cn_U(x)$ such that $f(y) - f(x) \in V$ for all $y \in U_x$ and $f \in \cf$. By compactness of $K$, there exists $\seqf{x_j} \subset K$ such that $K \subset \bigcup_{j = 1}^n U_{x_j}$. Since $B = \bigcup_{j = 1}^n \cf(x_j)$ is bounded, there exists $\lambda > 0$ such that $\lambda V \supset B$. In which case, 
    \[
    (\lambda + 1)V \supset B + V \supset \bigcup_{x \in K}^n \cf(x)
    \]

    (B2) $\Rightarrow$ (B1): By \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, $\bracsn{Df|f \in \cf}$ is also uniformly bounded on every compact set. Thus $\cf$ is equicontinuous. 

    (C1) $\Leftrightarrow$ (C2): By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}.
\end{proof}

\begin{definition}[Space of Holomorphic Functions Near a Set]
\label{definition:holomorphic-function-space-near}
    Let $E$ be a complete separated locally convex space over $\complex$ and $A \subset \complex$. Direct $\cn_{\complex}^o(A)$ under reverse inclusion, then the inductive limit
    \[
    H(A; E) = \varinjlim_{U \in \cn_{\complex}^o(A)} H(U; E)
    \]

    is the \textbf{space of holomorphic functions near} $A$, and is of type (LF). 
\end{definition}