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Housekeeping.

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Bokuan Li
2026-06-10 15:16:43 -04:00
parent 60c2144e9e
commit 1c211eac9a
3 changed files with 42 additions and 9 deletions
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src/fa/interpolation/complex.tex Normal file
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\section{The Complex Interpolation Method}
\label{section:complex-interpolation}

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src/fa/interpolation/functors.tex
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\begin{definition}[Compatible Couple]
\label{definition:compactible-couple}
Let $E_0, E_1, \mathcal{U}$ be topological vector spaces over $K \in \RC$ and $\iota_0 \in L(E_0; \mathcal{U})$ and $\iota_1 \in L(E_1; \mathcal{U})$ be continuous injections. Under the identification that $E_0$ and $E_1$ are subspaces of $\mathcal{U}$, the pair $(E_0, E_1)$ forms a \textbf{compatible couple} of topological vector spaces.
Let $E_0, E_1$ be topological vector spaces over $K \in \RC$, $\mathcal{U}$ be a separated topological vector space over $K$, and $\iota_0 \in L(E_0; \mathcal{U})$ and $\iota_1 \in L(E_1; \mathcal{U})$ be continuous injections. Under the identification that $E_0$ and $E_1$ are subspaces of $\mathcal{U}$, the pair $(E_0, E_1)$ forms a \textbf{compatible couple} of topological vector spaces.
\end{definition}
\begin{remark}
@@ -59,14 +59,14 @@
\normn{x - \sum_{n = 1}^Nx_n}_{E_0 + E_1} \le \sum_{n > N}\norm{y_n}_{E_0} + \sum_{n > N}\norm{z_n}_{E_1} \le 2\sum_{n > N}\norm{x_n}_{E_0 + E_1} \to 0
\]
as $N \to \infty$. Therefore $E_0 + E_1$ is also a Banach space.
as $N \to \infty$. By \autoref{lemma:banach-criterion}, $E_0 + E_1$ is also a Banach space.
\end{proof}
\begin{definition}[Category of Compatible Couples]
\label{definition:compatible-category}
Let $\catc$ be a subcategory of normed spaces over $K \in \RC$ and $(E_0, E_1)$ be a compatible couple, then $E_0 E_1$ are a \textbf{compatible couple in $\catc$} if $E_0, E_1, E_0 \cap E_1, E_0 + E_1 \in \catc$.
Let $(E_0, E_1)$ and $(F_0, F_1)$ be compatible couples in $\catc$ and $T \in L(E_0 + E_1, F_0 + F_1)$, then $T$ is a \textbf{morphism of compatible couples} if $T|_{E_0} \in \text{Mor}_{\catc}(E_0; F_0)$ and $T|_{E_1} \in \text{Mor}(E_1; F_1)$.
Let $(E_0, E_1)$ and $(F_0, F_1)$ be compatible couples in $\catc$ and $T \in L(E_0 + E_1, F_0 + F_1)$, then $T$ is a \textbf{morphism of compatible couples} if $T|_{E_0} \in \text{Mor}_{\catc}(E_0; F_0)$ and $T|_{E_1} \in \text{Mor}_{\catc}(E_1; F_1)$.
The collection $\catc_1$ of all compatible couples in $\catc$ equipped with the above definition of morphisms is the \textbf{category of compatible couples} in $\catc$.
\end{definition}
@@ -111,12 +111,6 @@
\end{definition}
\textit{"This is how things appeared in 1965. Fifteen years later, it was found that the number
of interpolation methods at our disposal is not large."} — \cite[Page vi, Footnote 3]{brudnyi1991interpolation}.