Examples of categorical adjunctions in analysis and differential geometry? In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and differential geometry (including Lie theory)? Obviously we have products, coproducts, etc., but those are common to a lot of categories we work with. Are there any examples more specific to analysis and differential geometry?
(Crosspost to Math Overflow)
 A: Here is a nice example: Consider the category $\mathsf{Ban}_1$ of Banach algebras with short linear maps. We have a functor $B_1 : \mathsf{Ban}_1 \to \mathsf{Set}$ which maps a Banach space to its unit ball. It has a left adjoint. This left adjoint maps a set $X$ to the Banach space $\ell^1(X)$ of summable functions on $X$. Since $\ell^1$ is left adjoint, we immediately see $\ell^1(X \sqcup Y) \cong \ell^1(X) \oplus \ell^1(Y)$.
Unitalization for rings can also be carried out for Banach algebras. This provides a functor $\mathsf{BanAlg} \to \mathsf{1BanAlg}$ which is left adjoint to the forgetful functor.
Another adjunction arises in C*-algebra theory: Consider the category $\mathsf{Alg}^*$ of $*$-algebras (over $\mathbb{C}$)  and the category $\mathsf{Top}$ of topological spaces. Then, the functor $\mathsf{Top} \to (\mathsf{Alg}^*)^{\mathrm{op}}$, $X \mapsto C(X,\mathbb{C})$  is left adjoint to the functor $\Phi : (\mathsf{Alg}^*)^{\mathrm{op}} \to \mathsf{Top}$ which maps a $*$-algebra $A$ to its character space $\Phi(A)$ whose underlying set is the set of $*$-homomorphisms $A \to \mathbb{C}$ and which is endowed with point-wise convergence. Every adjunction restricts to an equivalence of categories between its fixed points, which are determined by the classical Gelfand-Naimark Theorem: Compact Hausdorff spaces and commutative $C^*$-algebras.
