What's the adjoint for the evaluation operator?

Question

What's the adjoint for the evaluation operator, $A\in L([X\rightarrow Y],Y)$, where $Af=f(x)$ for some fixed $x\in X$? In case, there's any ambiguity, I'm looking for an operator $A^*$ such that $\langle Af,y\rangle_Y=\langle f(x),y\rangle_Y=\langle f,A^*y\rangle_{[X\rightarrow Y]}$. In case it's necessary, which it probably is for the inner product, it's fine to assume that $[X\rightarrow Y]$ is really a space of functions that's Lebesgue or Bochner integrable.

Answer 1

To the extent that this can be made concrete, you are looking at the reproducing kernel for your function space. The simplest example is when $\mathcal F$ is a Hilbert space of scalar-valued functions. Then the adjoint of point evaluation $E_{a}$ has one-dimensional domain, so it amounts to parametrizing a line: $$E_a^*(c) = ck_a,\quad \text{where } \ \langle f, k_a\rangle = f(a)\quad \forall f\in \mathcal F\tag1$$

More generally, $\mathcal F$ could be a Hilbert space of functions valued in another Hilbert space $\mathcal K$. The formula (1) still holds, except $c$ is now taken from $\mathcal K$.

Things are not as concrete in Banach spaces, since the adjoint takes values in $\mathcal F^*$. Reproducing kernel Banach spaces are also studied, but they are not as rich in structure. Without more specifics about $\mathcal F$, it's impossible to say what the elements of $\mathcal F^*$ will look like, and this is where the adjoint has to take its values. In this generality, one can only restate the general definition of an adjoint: each $c\in \mathcal K^*$ is mapped to a functional $\phi\in \mathcal F^*$ such that $ \phi(f) = \langle f(a), c\rangle $.