A Categorical Perspective on the Riesz Representation Theorem and Adjoint Maps

This blog is a rehash of the Riesz representation theorem in linear algebra, from a categorical point of view.

You will see why the adjoint map $T^{\ast }$$T^*$ is defined by $⟨Tv,w⟩=⟨v,T^{\ast }w⟩$$\langle Tv,w \rangle = \langle v,T^*w \rangle$.

To begin with, let us define a functor from the category of finite-dimensional inner product spaces to its dual category.

i.e.,

$$G:{\mathrm{Fin}}_{\mathbb{F}}\to {\mathrm{Fin}}_{\mathbb{F}}^{op}$$

For each object, $G:V⟼V$$G: V \longmapsto V$. For each morphism $T:V\to W$$T: V \to W$, $G:T⟼T^{\ast }$$G:T \longmapsto T^*$, where $T^{\ast }:W\to V$$T^*: W \to V$.

Another functor with which we are really familiar is the dual space functor ${\mathrm{Hom}}_{{\mathrm{Fin}}_{\mathbb{F}}}(-,\mathbb{F})$$\mathrm{Hom}_{\mathrm{Fin}_{\mathbb{F}}}(-,\mathbb{F})$, denoted as $F$$F$.

Riesz's representation theorem claims that there exists a natural isomorphism $\eta :G\to F$$\eta: G \to F$ such that the following diagram commutes:

$$\begin{array}{ccc}W& \stackrel{{\eta }_W}{\to }& W^{\ast }\\ T^{\ast }\downarrow & & \downarrow T^{\prime }& \\ V& \underset{{\eta }_V}{\overset{}{\to }}& V^{\ast }\end{array}$$

Indeed, the adjoint map $T^{\ast }$$T^*$ is defined by this diagram. $T^{\ast }$$T^*$ is the unique map that makes this diagram commute.

Firstly, let us prove the Riesz representation theorem.

Riesz representation theorem. Suppose $V\in \mathrm{Ob}({\mathrm{Fin}}_{\mathbb{F}})$$V \in \mathrm{Ob}(\mathrm{Fin}_{\mathbb{F}})$ and $\phi$$\varphi$ is a linear functional on $V$$V$. Then there exists a unique vector $v\in V$$v \in V$ such that

$$\phi (u)=⟨u,v⟩$$

for every $u\in V$$u \in V$.

Proof. First, we prove that there exists a $v\in V$$v \in V$ such that $\phi (u)=⟨u,v⟩$$\varphi(u) = \langle u, v \rangle$.

Let $e_1,e_2,\dots ,e_n$$e_1, e_2, \ldots, e_n$ be the standard orthogonal basis of $V$$V$. Then

$$\phi (u)=\phi (\sum _{i=1}^n⟨u,e_i⟩e_i)=\sum _{i=1}^n⟨u,e_i⟩\phi (e_i)=⟨u,\sum _{i=1}^n\overline{\phi (e_i)}{e}_i⟩$$

Hence,

$$v=\sum _{i=1}^n\overline{\phi (e_i)}{e}_i$$

To see $v$$v$ is unique, consider

$$\mathrm{\forall }u\in V,⟨u,v_1⟩=⟨u,v_2⟩⟹\mathrm{\forall }u\in V,⟨u,v_1-v_2⟩=0⟹v_1-v_2=0⟹v_1=v_2$$

Hence, the Riesz representation theorem gives us an isomorphism ${\eta }_V:V\to V^{\ast }$$\eta_V: V \to V^*$ for each $V$$V$. The isomorphism is natural.

To make the diagram commute,

$$\begin{array}{ccc}W& \stackrel{{\eta }_W}{\to }& W^{\ast }\\ T^{\ast }\downarrow & & \downarrow T^{\prime }& \\ V& \underset{{\eta }_V}{\overset{}{\to }}& V^{\ast }\end{array}$$

We have to define $T^{\ast }:={\eta }_V^{-1}\circ T^{\prime }\circ {\eta }_W$$T^* := \eta_V^{-1} \circ T' \circ \eta_W$.

From the diagram, we can see that ${\eta }_V\circ T^{\ast }=T^{\prime }\circ {\eta }_W$$\eta_V \circ T^* = T' \circ \eta_W$.

Putting a vector $w\in W$$w \in W$ on both sides, we can see that

$${\eta }_V\circ T^{\ast }(w)=T^{\prime }\circ {\eta }_W(w)⟺⟨-,T^{\ast }w⟩=T^{\prime }({\eta }_W(w))={\eta }_W(w)\circ T=⟨T(-),w⟩$$

Hence, we get the usual definition of the adjoint map.

$$⟨-,T^{\ast }w⟩=⟨T(-),w⟩$$

Readers might feel more familiar with this from

$$⟨v,T^{\ast }w⟩=⟨Tv,w⟩$$