Slight adjustments in known results.
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\item If $\fU \to x_0 \in \ol{D}$, then $f \in A(D)$, $\phi_{\fU}(f) = f(z_0)$.
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\item $\phi_{\fU}$ is a multiplicative linear functional on $H^\infty(D)$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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Let $f \in H^\infty(D)$, then by \autoref{proposition:imagefilterbase}, $f(\fU)$ is an ultrafilter base. Since $f$ is bounded, $f(\fU)$ converges to exactly one element of $\complex$. Hence the limit is well-defined.
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(2): By \autoref{proposition:operator-space-completeness}.
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(2): By \autoref{definition:multiplicative-linear-functional-space}.
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\end{proof}
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