Added the method of complex interpolation.

refs.bib

@@ -183,4 +183,17 @@
   author={Pietsch, Albrecht},
   year={2007},
   publisher={Springer}
+}
+
+@book{BerghInterpolation,
+  author    = {Bergh, Jöran and Löfström, Jörgen},
+  title     = {Interpolation Spaces: An Introduction},
+  series    = {Grundlehren der Mathematischen Wissenschaften},
+  volume    = {223},
+  publisher = {Springer-Verlag},
+  address   = {Berlin--New York},
+  year      = {1976},
+  pages     = {x+207},
+  isbn      = {3-540-07875-4},
+  mrnumber  = {0482275}
 }

src/fa/interpolation/complex.tex

@@ -0,0 +1,94 @@
+\section{The Complex Interpolation Method}
+\label{section:complex-interpolation}
+
+\begin{definition}[Calderón Space]
+\label{definition:calderon-space}
+    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 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}
+    
+    equipped with the norm
+    \[
+    \norm{f}_{\cf(E_0, E_1)} = \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_0}, \sup_{t \in \real}\norm{f(1 + it)}_{E_1}}
+    \]
+\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]{theorem:maximum-modulus-theorem}, \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]
+\label{definition:complex-interpolation-method}
+    Let $(E_0, E_1)$ be a compatible couple of Banach spaces over $\complex$, $\cf(E_0, E_1)$ be their \hyperref[Calderón space]{definition:calderon-space}, $\theta \in [0, 1]$, and
+    \[
+    [E_0, E_1]_\theta = \bracsn{f(\theta)| f \in \cf(E_0, E_1)}
+    \]
+
+    with the norm
+    \[
+    \norm{x}_{[E_0, E_1]_\theta} = \inf\bracsn{\norm{f}_{\cf(E_0, E_1)}| f \in \cf(E_0, E_1), x = f(\theta)}
+    \]
+
+    then:
+    \begin{enumerate}
+        \item $[E_0, E_1]_\theta$ is an intermediate Banach space between $E_0$ and $E_1$.
+        \item The mapping $C_\theta$ defined by $(E_0, E_1) \mapsto [E_0, E_1]_\theta$ is an interpolation functor of exact exponent $\theta$. 
+    \end{enumerate}
+
+    and functor $C_\theta$ is the \textbf{method of complex interpolation}. 
+\end{definition}
+\begin{proof}[Proof, {{\cite[Theorem 4.1.2]{BerghInterpolation}}}. ]
+    (1): Let $\seq{x_n} \subset [E_0, E_1]_\theta$ with $\sum_{n \in \natp}\norm{x_n}_{[E_0, E_1]_\theta} < \infty$, then there exists $\seq{f_n} \subset \cf(E_0, E_1)$ such that for each $n \in \natp$, $f_n(\theta) = x_n$ and $\norm{f_n}_{\cf(E_0, E_1)} \le 2\norm{x_n}_{[E_0, E_1]_\theta}$. Since $\cf(E_0, E_1)$ is complete, there exists $f \in \cf(E_0, E_1)$ such that $f = \sum_{n = 1}^\infty f_n$. Let $x = f(\theta)$, then since $\sum_{n = 1}^N f_n \to f$ in $\cf(E_0, E_1)$ as $N \to \infty$, $\sum_{n = N}^\infty x_n \to x$ in $[E_0, E_1]_\theta$ as $N \to \infty$. Therefore $[E_0, E_1]_\theta$ is a Banach space by \autoref{lemma:banach-criterion}. 
+
+    For any $x \in E_0 \cap E_1$ and $\delta > 0$, let $f_\delta(z) = x_0 e^{(z - \theta)^2}$, then $f_\delta \in \cf(E_0, E_1)$ with $\norm{f_\delta}_{\cf(E_0, E_1)} \le e^\delta\norm{x}_{E_0 \cap E_1}$. Thus $x \in [E_0, E_1]_\theta$ with
+    \[
+    \norm{x}_{[E_0, E_1]_\theta} \le \norm{f}_{\cf(E_0, E_1)} \le e^\delta\norm{x}_{E_0 \cap E_1}
+    \]
+
+    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}, 
+    \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}} \\ 
+    &\le  \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_1}, \sup_{t \in \real}\norm{f(1 + it)}_{E_1}} = \norm{f}_{\cf(E_0, E_1)}
+    \end{align*}
+
+    so $[E_0, E_1]_\theta$ is continuously embedded in $E_0 + E_1$. 
+    
+
+    (2): Let $(F_0, F_1)$ be a compatible couple of Banach spaces over $\complex$, and $T \in L(E_0 + E_1; F_0 + F_1)$ such that $T|_{E_0} \in L(E_0; F_0)$ and $T|_{E_1} \in L(E_1; F_1)$. Let $x \in [E_0, E_1]_\theta$ and $f \in \cf(E_0, E_1)$ such that $f(\theta) = x$. For each $z \in \bracs{y \in \complex| \text{Re}(y) \in [0, 1]}$, let
+    \[
+    g(z) = \norm{T}_{L(E_0; F_0)}^{z - 1} \norm{T}_{L(E_1; F_1)}^{-z} \cdot T \circ f(z)
+    \]
+
+    then for each $t \in \real$, 
+    \begin{align*}
+        \norm{g(it)}_{F_0} &= \norm{T}_{L(E_0; F_0)}^{-1} \cdot \norm{T \circ f(it)}_{F_0} \le \norm{f(it)}_{E_0} \\
+        \norm{g(1 + it)}_{F_0} &= \norm{T}_{L(E_1; F_1)}^{-1} \cdot \norm{T \circ f(1 + it)}_{F_1} \le \norm{f(1 + it)}_{E_1}
+    \end{align*}
+    
+    so $g \in \cf(F_0, F_1)$ with $\norm{g}_{\cf(F_0, F_1)} \le \norm{f}_{\cf(E_0, E_1)}$. Thus
+    \begin{align*}
+    g(\theta) &= \norm{T}_{L(E_0; F_0)}^{\theta - 1} \norm{T}_{L(E_1; F_1)}^{-\theta} \cdot T \circ f(\theta) \\
+    &= \norm{T}_{L(E_0; F_0)}^{\theta - 1} \norm{T}_{L(E_1; F_1)}^{-\theta} \cdot Tx
+    \end{align*}
+
+    and $Tx \in [F_0, F_1]_\theta$ with
+    \begin{align*}
+        \norm{Tx}_{[F_0, F_1]_\theta} &\le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta}\norm{g}_{\cf(F_0, F_1)} \\
+        &\le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta} \norm{f}_{\cf(E_0, E_1)}
+    \end{align*}
+
+    Since the above holds for all $f \in \cf(E_0, E_1)$ with $f(\theta) = x$, 
+    \[
+    \norm{Tx}_{[F_0, F_1]_\theta} \le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta} \norm{x}_{[E_0, E_1]_{\theta}}
+    \]
+\end{proof}
+
+

src/fa/interpolation/functors.tex

@@ -5,7 +5,7 @@
 
 \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}
@@ -104,19 +104,13 @@
 \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 $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$
     \[
-    \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; E_1)}^\theta\norm{T}_{L(F_0; F_1)}^{1 - \theta}
+    \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; F_0)}^{1 - \theta}\norm{T}_{L(E_1; F_1)}^{\theta}
     \]
 
     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}. 
 

src/fa/interpolation/index.tex

@@ -2,5 +2,6 @@
 \label{chap:interpolation}
 
 \input{./functors.tex}
+\input{./complex.tex}
 
 

src/fa/norm/normed.tex

@@ -10,8 +10,44 @@
         \item[(SN2)] For any $x \in E$ and $\lambda \in K$, $\norm{\lambda x}_E = \abs{\lambda} \norm{x}_E$.
         \item[(SN3)] For any $x, y \in E$, $\norm{x + y}_E \le \norm{x}_E + \norm{y}_E$.
     \end{enumerate}
+
+    In which case, the pair $(E, \norm{\cdot}_E)$ is a \textbf{normed vector space} over $K$. 
 \end{definition}
 
+\begin{definition}[Banach Space]
+\label{definition:banach-space}
+    Let $E$ be a normed vector space over $K \in \RC$, then $E$ is a \textbf{Banach space} if it is complete. 
+\end{definition}
+
+\begin{lemma}
+\label{lemma:banach-criterion}
+    Let $E$ be a normed vector space, then the following are equivalent:
+    \begin{enumerate}
+        \item $E$ is a Banach space.
+        \item For each $\seq{x_n} \subset E$ with $\sum_{n \in \natp}\norm{x_n}_E < \infty$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (2) $\Rightarrow$ (1): Let $\seq{x_n} \subset E$ be a Cauchy sequence, then there exists a subsequence $\seq{n_k}$ such that $\norm{x_{n_{k+1}} - x_{n_{k}}}_E < 2^{-k}$ for all $k \in \natp$. For each $k \in \natp$, let $y_k = x_{n_{k+1}} - x_{n_{k}}$, then there exists $y \in E$ such that $y = \sum_{k = 1}^\infty y_k$. Let $\eps > 0$, then there exists $K \in \natp$ such that:
+    \begin{enumerate}[label=(\alph*)]
+        \item $\norm{y - \sum_{k = 1}^{K-1} y_k}_E < \eps$. 
+        \item For each $n \ge n_K$, $\norm{x_n - x_{n_K}}_E < \eps$. 
+    \end{enumerate}
+
+    In which case, for every $n \ge n_K$, 
+    \begin{align*}
+        \norm{x_n - (y + x_{n_1})}_E &< \norm{x_{n_K} - (y + x_{n_1})}_E + \eps \\
+        &= \norm{y - \sum_{k = 1}^{K-1} (x_{n_{k+1}} - x_{n_{k}})}_E + \eps \\
+        &= \norm{y - \sum_{k = 1}^{K-1} y_k}_E + \epsilon < 2\eps
+    \end{align*}
+
+    Therefore $x_n \to y + x_{n_1}$ as $n \to \infty$. 
+    
+    
+\end{proof}
+
+
+
 \begin{proposition}
 \label{proposition:norm-criterion}
     Let $E$ be a separated TVS over $K \in \RC$, then the following are equivalent:

src/fa/notation.tex

@@ -49,5 +49,9 @@
     $\mu_G$ & Lebesgue-Stieltjes measure associated with $G$. & \autoref{definition:riemann-lebesgue-stieltjes} \\
     $\int_\gamma f$, $\int_\gamma f(z)dz$ & Path integral of $f$ with respect to $\gamma$. & \autoref{definition:path-integral} \\
     $PI([a, b], \gamma; E)$ & Space of path integrable functions with respect to $\gamma$. & \autoref{definition:path-integral}
+    % ---- Interpolation Spaces ---- \\
+    $\catc_1$ & Category of compatible couples in $\catc$. & \autoref{definition:compatible-category} \\
+    $\cf(E_0, E_1)$ & Calderón space of $(E_0, E_1)$ & \autoref{definition:calderon-space} \\
+    $[E_0, E_1]_\theta$ & Complex interpolation space of exponent $\theta$ for the couple $(E_0, E_1)$. & \autoref{definition:complex-interpolation-method}
 \end{tabular}