I read (p. 485 here) that the Fréchet derivative of norm (non-linear) functional $p:H\to\mathbb{R}$, $x\mapsto\|x\|$ is $\frac{x}{\|x\|}$ for all $x\ne 0$, which I think to be intended as the linear functional, which is what the Fréchet derivative $p'(x)\in \mathscr{L}(H,\mathbb{R})$ should be, defined by $h\mapsto\langle\frac{x}{\|x\|},h\rangle$. I suspect $H$ is intended to be a real Hilbert space, although Kolmogorov-Fomin's says nothing about the scalar field.
As always, before asking here, I searched the Internet, but only find this page from this very site. Can anybody point to a proof of $p'(x)h=\langle\frac{x}{\|x\|},h\rangle$? I $\infty$-ly thank you!