\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}