Slight adjustments in known results.

This commit is contained in:
Bokuan Li
2026-06-03 14:37:13 -04:00
parent 2c1169e55a
commit 56f3ae37f7
2 changed files with 10 additions and 6 deletions

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@@ -23,12 +23,11 @@
\item If $\fU \to x_0 \in \ol{D}$, then $f \in A(D)$, $\phi_{\fU}(f) = f(z_0)$.
\item $\phi_{\fU}$ is a multiplicative linear functional on $H^\infty(D)$.
\end{enumerate}
\end{proposition}
\begin{proof}
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.
(2): By \autoref{proposition:operator-space-completeness}.
(2): By \autoref{definition:multiplicative-linear-functional-space}.
\end{proof}