-----
> [!proposition] Proposition. ([[dual of quotient is annihilator]] - algebraic)
> If $W \leq V$ is an inclusion of [[vector space|vector spaces]], then $W^{*}= \frac{V^{*}}{W^{\circ}} \text{ and } \left( \frac{V}{W} \right)^{*} = W^{\circ},$
where $(\cdot)^{*}$ denotes (algebraic) [[dual vector space|dual]] and $(\cdot)^{\circ}$ denotes [[orthogonal complement|annihilator]]. We use equal signs to emphasize the [[natural transformation|naturality]] of these [[linear map|linear]] [[isomorphism|isomorphisms]].
>
The second isomorphism holds more generally in [[module|the]] [[category]] $R$-$\mathsf{Mod}$. The first does not!
^proposition
> [!proposition] Proposition. ([[dual of quotient is annihilator]] - topological; Theorem 5.18)
> The identity $\left( \frac{V}{W} \right)^{*}=W^{\circ}$
> cleanly transfers to [[topological vector space|TVS]] setting with [[continuous]] [[dual vector space|duals]]. Note than in (say) $\mathsf{Norm_{T}}$, one will [[characterization of quotienting a (semi)normed vector space|as usual]] want $W$ [[closed set|closed]] to get an earnest [[norm]] on the [[quotient module|quotient]]. The other identity $W^{*}=\frac{V^{*}}{W^{\circ}}$
> holds for [[norm|normed]] (more generally [[locally convex space|locally convex]]) spaces, but the proof needs [[Hahn-Banach Extension Theorem|Hahn-Banach]] (it uses Theorem 5.16) and indeed can fail in $\mathsf{TVS}$.
If $V$ is finite-dimensional, then taking duals on both sides of the first identity and using $W^{* *}=W$ and the second identity, we get a further [[natural transformation|natural isomorphism]] $W=$
then we get additional [[natural transformation|natural isomorphisms]] $W=\frac{V}{}$ $$
> [!proof]+ Proof. ([[dual of quotient is annihilator]])
> **Isomorphism 1.** Let $\iota:W \hookrightarrow V$ denote [[inclusion map|inclusion]]. The [[dual map]] $\iota^{*}$ is restriction to $W$: $\begin{align}
> V^{*} &\xrightarrow{\iota^{*} }W^{*} \\
> \varphi & \mapsto \varphi |_{W}.
> \end{align}$
> $\iota^{*}$ is a [[surjection]] since $\iota$ is an [[injection]]. We have $\operatorname{ker }\iota^{*}=\{ \varphi \in V^{*}: \varphi |_{W} \equiv 0 \}=W^{\circ}.$
> [[characterization of quotienting a module|By]] [[first isomorphism theorem for modules|the FIT]], we conclude $W^{*} \cong \frac{V^{*}}{W^{\circ}}$.
>
>
>
> **Isomorphism 2.** Any element $\overline{\varphi} \in (\frac{V}{W})^{*}$ [[characterization of quotienting a module|induces]] a unique $\varphi \in V^{*}$ such that the [[diagram]] commutes:
>
> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADUQBfU9TXfIRRkAjFVqMWbdgHoA6t14gM2PASIjy4+s1aIQAa27iYUAObwioAGYAnCAFskZEDghIRPG-aeIXbpAAmah0pfQAdcJgADyw4HDgAAgBCRMj6WzQACyxFb0cPagDEYIldNkiIGhhbBiwwGGB0ukycrhBqBjoAIxgGAAV+NSEQWywzLJxjLiA
> \begin{tikzcd}
> V \arrow[d] \arrow[r, "\exists ! \varphi"] & k \\
> V/W \arrow[ru, "\overline{\varphi}"'] &
> \end{tikzcd}
> \end{document}
> ```
>
> This situation is encoded by the [[linear map|linear map]] $\begin{align}
> \left( \frac{V}{W} \right)^{*} &\xrightarrow{\Psi} W^{\circ} \\
> \overline{\varphi} & \mapsto \varphi, \text{ where }\varphi(v)= \overline{\varphi}(v + W)
> \end{align}$
> where we've noted $\varphi \in W^{\circ}$ when corestricting $V^{*}$ to $W^{\circ}$. By the '$\ex ! \varphi