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Fixed gap in maximum modulus strip.

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This commit is contained in:
Bokuan Li
2026-06-26 00:44:31 -04:00
parent 91752ee561
commit fbdf280f11
2 changed files with 34 additions and 5 deletions
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8
src/fa/interpolation/complex.tex
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@@ -6,7 +6,7 @@
Let $S = \bracs{z \in \complex| \text{Re}(z) \in (0, 1)}$ and $(E_0, E_1)$ be a compatible couple of Banach spaces over $\complex$, then the \textbf{Calderón space} $\cf(E_0, E_1)$ is the Banach space of functions $f: \ol S \to E_0 + E_1$ such that:
\begin{enumerate}
\item $f$ is holomorphic on $S$.
\item $f$ is continuous on $\ol S$.
\item $f$ is bounded and continuous on $\ol S$.
\item For each $t \in \real$, $f(it) \in E_0$, and $\lim_{|t| \to \infty}\norm{f(it)}_{E_0} = 0$.
\item For each $t \in \real$, $f(1 + it) \in E_1$, and $\lim_{|t| \to \infty}\norm{f(1 + it)}_{E_1} = 0$.
\end{enumerate}
@@ -17,9 +17,9 @@
\]
\end{definition}
\begin{proof}
By the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem} applied to $f$ as a function in $H(S; E_0 + E_1)$, $\norm{\cdot}_{\cf(E_0, E_1)}$ is a norm.
By the \hyperref[Maximum Modulus Theorem]{lemma:maximum-modulus-strip} applied to $f$ as a function in $H(S; E_0 + E_1)$, $\norm{\cdot}_{\cf(E_0, E_1)}$ is a norm.
By the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem}, \autoref{proposition:holomorphic-complete}, and \autoref{proposition:uniform-limit-continuous}, $\cf(E_0, E_1)$ is complete.
By the \hyperref[Maximum Modulus Theorem]{lemma:maximum-modulus-strip}, \autoref{proposition:holomorphic-complete}, and \autoref{proposition:uniform-limit-continuous}, $\cf(E_0, E_1)$ is complete.
\end{proof}
\begin{definition}[The Complex Interpolation Method]
@@ -52,7 +52,7 @@
As the above holds for all $\delta > 0$, $E_0 \cap E_1$ is continuously embedded in $[E_0, E_1]_\theta$.
Let $x \in [E_0, E_1]_\theta$ and $f \in \cf(E_0, E_1)$ with $f(\theta) = x$, then by the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem},
Let $x \in [E_0, E_1]_\theta$ and $f \in \cf(E_0, E_1)$ with $f(\theta) = x$, then by the \hyperref[Maximum Modulus Theorem]{lemma:maximum-modulus-strip},
\begin{align*}
\norm{x}_{E_0 + E_1} &= \norm{f(\theta)}_{E_0 + E_1} \\
&\le \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_0 + E_1}, \sup_{t \in \real}\norm{f(1 + it)}_{E_0 + E_1}} \\