\section{Compatible Couples}
\label{section:interpolation-functors}

\textit{"In the presence of so many different interpolation methods it seemed timely to study the general structure of all possible methods: to determine all of them and to analyze the properties which are common to all."} — \cite[Page 51]{aronszajn1964interpolation}. 

\begin{definition}[Compatible Couple]
\label{definition:compactible-couple}
    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}
\label{remark:compatible-couple}
    The structure of the compatible couple depends on the common space and the inclusion maps. As such, the couple $(E_0, E_1)$ always implicitly carries the common space and the injections. 
\end{remark}


\begin{definition}[Sum and Intersection Spaces]
\label{definition:sum-intersection-spaces}
    Let $(E_0, E_1)$ be a compatible couple of topological vector spaces over $K \in \RC$, then $E_0 \cap E_1$ is their \textbf{intersection space}, and $E_0 + E_1$ is their \textbf{sum space}. 
\end{definition}

\begin{definition}
\label{definition:sum-intersection-norm}
    Let $(E_0, E_1)$ be a compatible couple of normed vector spaces over $K \in \RC$, then:
    \begin{enumerate}
        \item $E_0 \cap E_1$ is a normed space under the norm
        \[
        \norm{\cdot}_{E_0 \cap E_1}: E_0 \cap E_1 \to [0, \infty) \quad x \mapsto \max(\norm{x}_{E_0}, \norm{x}_{E_1})
        \]
        \item $E_0 + E_1$ is a normed space under the norm
        \[
        \norm{\cdot}_{E_0 + E_1}: E_0 + E_1 \to [0, \infty)
        \]

        with
        \[
        x \mapsto \inf\bracsn{\norm{x_0}_{E_0} + \norm{x_1}_{E_1}|x_0 \in E_0, x_1 \in E_1, x = x_0 + x_1}
        \]

        \item If $E_0$ and $E_1$ are Banach spaces, then $E_0 \cap E_1$ and $E_0 + E_1$ are also Banach spaces. 
    \end{enumerate}

    The norms on $E_0 \cap E_1$ and $E_0 + E_1$ defined above are the \textbf{intersection} and \textbf{sum} norms of the couple, respectively. 
\end{definition}
\begin{proof}
    (2): Let $x, y \in E_0 + E_1$, $x_0, y_0 \in E_0$, $x_1, y_1 \in E_1$ such that $x = x_0 + x_1$ and $y = y_0 + y_1$, then
    \begin{align}
    \norm{x + y}_{E_0 + E_1} &\le \norm{x_0 + y_0}_{E_0} + \norm{x_1 + y_1}_{E_1} \\
    &\le (\norm{x_0}_{E_0} + \norm{x_1}_{E_1}) + (\norm{y_0}_{E_0} + \norm{y_1}_{E_1})
    \end{align}

    As this holds for all choices of $x_0, y_0 \in E_0$ and $x_1, y_1 \in E_1$, 
    \[
    \norm{x + y}_{E_0 + E_1} \le \norm{x}_{E_0 + E_1} + \norm{y}_{E_0 + E_1}
    \]

    (3): Let $\seq{x_n} \subset E_0 + E_1$ such that $\sum_{n \in \natp}\norm{x_n}_{E_0 + E_1} < \infty$. For each $n \in \natp$, let $y_n \in E_0$ and $z_n \in E_1$ with $x_n = y_n + z_n$ and $\norm{y_n}_{E_0} + \norm{z_n}_{E_1} \le 2\norm{x_n}_{E_0 + E_1}$. Since $E_0$ and $E_1$ are both complete, $y = \sum_{n = 1}^\infty y_n$ exists in $E_0$ and $z = \sum_{n = 1}^\infty z_n$ exists in $E_1$. Let $x = y + z$, then for each $N \in \natp$, 
    \[
    \normn{x - \sum_{n = 1}^Nx_n}_{E_0 + E_1} \le \sum_{n > N}\norm{y_n}_{E_0} + \sum_{n > N}\norm{z_n}_{E_1} \le 2\sum_{n > N}\norm{x_n}_{E_0 + E_1} \to 0
    \]

    as $N \to \infty$. 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 $T|_{E_1} \in \text{Mor}_{\catc}(E_1; F_1)$. 

    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}

\begin{definition}[Intermediate Space]
\label{definition:intermediate-space}
    Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $(E_0, E_1) \in \catc_1$ be a compatible couple in $\catc$, and $E \in \catc$, then $E$ is an \textbf{intermediate space} between $E_0$ and $E_1$ if there exists continuous inclusions
    \[
    \xymatrix{
    E_0 \cap E_1 \ar@{->}[r] & E \ar@{->}[r] & E_0 + E_1
    }
    \]
\end{definition}

\begin{definition}[Interpolation Spaces]
\label{definition:interpolation-spaces}
    Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $(E_0, E_1), (F_0, F_1) \in \catc_1$ be a compatible couple in $\catc$, and $E, F \in \catc$, then $E$ and $F$ are \textbf{interpolation spaces} with respect to $(E_0, E_1)$ and $(F_0, F_1)$ if
    \begin{enumerate}
        \item $E$ is an intermediate space between $E_0$ and $E_1$. 
        \item $F$ is an intermediate space between $F_0$ and $F_1$. 
        \item For any $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0; F_1))$, $T|_{E} \in \text{Mor}_{\catc}(E; F)$. 
    \end{enumerate}
\end{definition}

\begin{definition}[Interpolation Functor]
\label{definition:interpolation-functor}
    Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_1$ be its categories of compatible couples, and $F: \catc_1 \to \catc$ be a functor, then $F$ is an \textbf{interpolation functor} if for every $(E_0, E_1), (F_0, F_1) \in \catc_1$, 
    \begin{enumerate}
        \item $F((E_0, E_1))$ and $F((F_0, F_1))$ are interpolation spaces with respect to $(E_0, E_1)$ and $(F_0, F_1)$. 
        \item For each $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$, $F(T) = T|_{F((E_0, E_1))}$. 
    \end{enumerate}
\end{definition}

\begin{definition}[Interpolation Exponent]
\label{definition:interpolation-functor-exponent}
    Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_1$ be its categories of compatible couples, $F: \catc_1 \to \catc$ be an interpolation functor, and $\theta \in [0, 1]$, then $F$ is \textbf{of exponent $\theta$} if there exists $C \ge 0$ such that for every $(E_0, E_1), (F_0, F_1) \in \catc_1$ and $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$
    \[
    \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; F_0)}^{1 - \theta}\norm{T}_{L(E_1; F_1)}^{\theta}
    \]

    If $C = 1$, then $F$ is \textbf{of exact exponent $\theta$}. 
\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}. 

The above quotes are taken from \cite[Page 427]{PietschHistory}.