diff --git a/src/dg/complex/zero.tex b/src/dg/complex/zero.tex index 3174884..f3540e5 100644 --- a/src/dg/complex/zero.tex +++ b/src/dg/complex/zero.tex @@ -105,7 +105,7 @@ then for each $s \in [0, 1]$, $M(s) \le M(0)^s M(1)^{1-s}$. \end{lemma} -\begin{proof} +\begin{proof}[Proof, {{\cite[Theorem VI.3.7]{ConwayComplex}}}. ] Assume without loss of generality that $M(0), M(1) > 0$. Let \[ g: \complex \to \complex \quad z \mapsto M(0)^z M(1)^{1 - z} diff --git a/src/fa/index.tex b/src/fa/index.tex index adeace0..d604623 100644 --- a/src/fa/index.tex +++ b/src/fa/index.tex @@ -9,4 +9,5 @@ \input{./lp/index.tex} \input{./order/index.tex} \input{./duality/index.tex} +\input{./interpolation/index.tex} \input{./notation.tex} diff --git a/src/fa/interpolation/functors.tex b/src/fa/interpolation/functors.tex index 459335a..cc2c835 100644 --- a/src/fa/interpolation/functors.tex +++ b/src/fa/interpolation/functors.tex @@ -1,8 +1,122 @@ -\section{Interpolation Functors} +\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 $E_0, E_1, \mathcal{U}$ 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. +\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 $N \to \infty$. Therefore $E_0 + E_1$ 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 $T|_{E_1} \in \text{Mor}(E_1; F_1)$. + + 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 $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$ + \[ + \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; E_1)}^\theta\norm{T}_{L(F_0; F_1)}^{1 - \theta} + \] + + 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}.