Problem: Let $a,b\in \mathbb{R} $. We consider a real linear space $V$ of polynomials on $[a, b]$ of degree no larger than $2023$ with the scalar product
\[
\left\langle f,g \right\rangle \coloneqq \int_{a}^b f(t) g(t) \mathrm{d} t.
\]
Let a real valued function $k(s,t)$ be continuous for $s \in [a,b]$ and $t \in [a,b]$. Let us define the linear map $F: V \to V$ by
\[
(F(f))(t) = \int _a^b k(s,t)f(s) \mathrm{d} s,\ t \in [a,b].
\]
Determine an explicit expression for $F^{\star}$, the adjoint of $F$.
Solution: Note that $V$ is a finite dimensional vector space. Let $f,g \in V$. We recall that a linear operator $T$ on an inner product space $V$ is said to have an adjoint operator $T^\star $ on $V$ if $\left\langle T(u), v \right\rangle = \left\langle u, T^\star ,v \right\rangle $, for all $u,v \in V$. Note that
\begin{align*}
\left\langle g, F(f) \right\rangle & = \int_a^b g(t) (F(f))(t)~\mathrm{d} t \\
& = \int_a^b g(t) \left( \int _a^b k(s,t)f(s) \mathrm{d} s \right) \mathrm{d} t\\
& = \int_a^b \int_a^b k(s,t)f(s) g(t)~\mathrm{d} s ~ \mathrm{d} t \\
& = \int _a^b \left( \int _a^b K(s,t) g(t)~\mathrm{d} t \right) f(s)~\mathrm{d} s,
\end{align*}
where for the last equality we used the Fubini's theorem to switch the integration.
If we define,
\[
F^\star : V \to V, (F^*(f))(t) \coloneqq \int _a^b k(s,t)g(t)~\mathrm{d} t,\ t \in [a,b],
\]
then we see that for any $f,g \in V$ we have
\[
\left\langle g, F(f) \right\rangle = \left\langle F^\star (g),f \right\rangle .
\]
Therefore, $F^\star $ is the adjoint of $F$.