Posted on 8th September 2023

This is a follow-on to an older post prompted by a comment from Yemon Choi (an old friend who is now a colleague). There are of course many examples of monotone Galois connections in Functional Analysis. Here is one example which I like.

Let \( E \) be a Banach space. Let \( \mathcal X \) be the collection of all linear subspaces of \( E \), ordered by inclusion. Let \( \mathcal Y \) be the collection of all linear subspaces of \( E^* \) (the dual space of all continuous linear functionals) ordered by reverse inclusion.

Given \( F\in\mathcal X \) we define \[ \mathcal L(F) = \{ f\in E^* : f(x)=0 \ (x\in F) \} \in \mathcal Y, \] the annihilator of \( F \), easily seen to be a subspace. If \( F_1 \leq F_2 \) then \( F_1 \subseteq F_2 \) and so \( \mathcal L(F_1) \supseteq \mathcal L(F_2) \) (if \( f \) vanishes on \( F_2 \) it will vanish on \( F_1 \)). So \( \mathcal L(F_1) \leq \mathcal L(F_2) \) and \( \mathcal L \) is order-preserving.

Similarly given \( F\in\mathcal Y \) define \[ \mathcal R(F) = \{ x\in E : f(x)=0 \ (f\in F) \}\in \mathcal X, \] the pre-annihilator of \( F \). This is again an order-preserving map.

Let \( F\in\mathcal X \) and \( G\in\mathcal Y \). Then \[ \begin{split} & \mathcal L(F) \leq G \\ \Leftrightarrow\quad & \mathcal L(F) \supseteq G \\ \Leftrightarrow\quad & f\in G \implies f(x)=0 \ (x\in F) \\ \Leftrightarrow\quad & x\in F \implies f(x)=0 \ (f\in G) \\ \Leftrightarrow\quad & F \subseteq \mathcal R(G) \\ \Leftrightarrow\quad & F \leq \mathcal R(G). \end{split} \] Hence we have the required condition.

If we run the abstract argument, we obtain a bijection between the images of \( \mathcal R \) and \( \mathcal L \). Here some non-trivial Functional Analysis appears in the guise of the Hahn-Banach Theorem. The image of \( \mathcal L \) is the collection of all weak\( ^* \)-closed subspaces of \( E^\ast \), while the image of \( \mathcal R \) is the collection of all (norm) closed subspaces of \( E \). This bijection is of course well-known, but it's fun to see it appear here.