Added elementary properties of adjoint maps.

src/fa/duality/adjoint.tex

@@ -0,0 +1,75 @@
+\section{Adjoint Maps}
+\label{section:adjoint-maps}
+
+\begin{definition}[Adjoint Map]
+\label{definition:adjoint-map}
+    Let $E, F$ be vector spaces over a field $K$, and $T \in \hom(E; F)$ be a linear map, then the mapping
+    \[
+    T^*: F^* \to E^* \quad \dpn{x, T^*\phi}{E} = \dpn{Tx, \phi}{F}
+    \]
+
+    is the \textbf{algebraic adjoint} of $T$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:adjoint-weak-continuous}
+    Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T \in \hom(E; G)$, then the following are equivalent:
+    \begin{enumerate}
+        \item $T$ is $\sigma(E, F)$-$\sigma(G, H)$ continuous. 
+        \item $T^*(H) \subset F$. 
+    \end{enumerate}
+
+    If the above holds, then
+    \begin{enumerate}[start=2]
+        \item $T^*|_{H}$ is $\sigma(H, G)$-$\sigma(F, E)$ continuous. 
+        \item $T^{**} = T$. 
+    \end{enumerate}
+
+    and the restriction of $T^*$ to $H$ is the \textbf{adjoint} of $T$ with respect to $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$. 
+\end{proposition}
+% Proof omitted due to obviousness. 
+
+\begin{proposition}
+\label{proposition:adjoint-polar-gymnastics}
+    Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$, $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, $A \subset E$, and $B \subset G$, then:
+    \begin{enumerate}
+        \item $T(A)^\circ = (T^{*})^{-1}(A^\circ)$.
+        \item If $T(A) \subset B$, then $T^{*}(B^\circ) \subset A^\circ$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): 
+    \begin{align*}
+        T(A)^\circ &= \bracsn{\phi \in H| \text{Re}\dpn{Tx, \phi}{\mu} \le 1 \forall x \in A} \\
+        &= \bracsn{\phi \in H| \text{Re}\dpn{x, T^*\phi}{\lambda} \le 1 \forall x \in A} = (T^{*})^{-1}(A^\circ)
+    \end{align*}
+
+    (2): 
+    \begin{align*}
+        T^*(B^\circ) &= T^*(\bracs{\phi \in H| \text{Re}\dpn{y, \phi}{\mu} \le 1 \forall y \in B}) \\
+        &\subset T^*(\bracs{\phi \in H| \text{Re}\dpn{y, \phi}{\mu} \le 1 \forall y \in T(A)}) \\
+        &= T^*(\bracs{\phi \in H| \text{Re}\dpn{Tx, \phi}{\mu} \le 1 \forall x \in A})\\
+        &= T^*(\bracs{\phi \in H| \text{Re}\dpn{x, T^*\phi}{\lambda} \le 1 \forall x \in A}) \subset A^\circ
+    \end{align*}
+\end{proof}
+
+\begin{corollary}
+\label{corollary:adjoint-kernel-gymnastics}
+    Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, then:
+    \begin{enumerate}
+        \item $\ker(T^*) = T(E)^\perp = \bracs{\phi \in H| \dpn{y, \phi}{\mu} = 0 \forall y \in T(E)}$. 
+        \item $T^*$ is injective if and only if $T(E)$ is $\sigma(G, H)$-dense in $G$. 
+    \end{enumerate}
+\end{corollary}
+
+\begin{proposition}
+\label{proposition:adjoint-continuity}
+    Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$, $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, $\sigma \subset 2^E$ be a saturated ideal of $\sigma(E, F)$-bounded sets, $\tau \subset 2^G$ be a saturated ideal of $\sigma(G, H)$-bounded sets, then the following are equivalent:
+    \begin{enumerate}
+        \item $T^*$ is continuous with respect to the $\tau$-uniform topology on $H$ and the $\sigma$-uniform topology on $F$. 
+        \item $T(\sigma) \subset \tau$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): Let $A \in \sigma$, then there exists $B \in \tau$ such that $T^*\phi(A) \subset \ol{B_K(0, 1)}$ for all $\phi \in H$ with $\phi(B) \subset \ol{B_K(0, 1)}$. In which case, $T^*(B^\circ) \subset A^\circ$.  Assume without loss of generality that $A$ and $B$ are convex, circled, and closed. By \autoref{proposition:adjoint-polar-gymnastics} applied to $T^*$ and the \hyperref[Bipolar theorem]{theorem:bipolar}, $T(A) \subset B$. Therefore $T(\sigma) \subset \tau$. 
+\end{proof}

src/fa/duality/index.tex

@@ -4,5 +4,6 @@
 \input{./definitions.tex}
 \input{./polar.tex}
 \input{./mackey.tex}
+\input{./adjoint.tex}
 
 

src/op/vn/topologies.tex

@@ -13,7 +13,7 @@ Depending on the topology placed on $H \otimes H$, and the corresponding complet
     Let $H$ be a complex Hilbert space, then the dual of $H \wh \otimes_\pi H$ is $B(H)$, and the $\sigma(B(H), H \wh \otimes_\pi H)$-topology is the \textbf{ultraweak}/\textbf{$\sigma$-weak} topology on $B(H)$. 
 \end{definition}
 \begin{proof}
-    By \autoref{proposition:projective-tensor-product-dual}. 
+    By \autoref{proposition:projective-tensor-product-dual} and the \hyperref[Riesz Representation Theorem]{theorem:riesz-hilbert}. 
 \end{proof}
 
 Seeing that $B(H)$ is a dual Banach space, the following fact is immediate: