Frechet derivative of composition

Question

Let $f:\mathbb{R} \to \mathbb{R}$ be differentiable at each $x\in \mathbb{R}$ and let $\mathcal{H} \subset \{\mathbb{R}^n\to \mathbb{R}\}$ be a reproducing kernel Hilbert space.

Is it true that $f_x:\mathcal{H} \to \mathbb{R}, f_x(h) := f(h(x)) $, for some $x\in\mathbb{R} $, is Frechet differentiable in each $h\in \mathcal{H}$?

Answer 1

Since $\mathcal{H} $ is an RKHS, the (linear) evaluation functional $B_x(h)=h(x) $ is bounded and it's Frechet derivative therefore exists. As a consequence, the composition $(f'\circ B_x)(h) B_x'(h)$ also exists, which equals the desired quantity $(f\circ B_x)'(h)=f_x'(h)$ by the chain rule.