Is the Integral Operator $L_p(\mathbb{R}^n)\ni\varphi\mapsto \int_{\mathbb{R}^n} \varphi (x) g(x) dx, g\in L_{p'}(\mathbb{R}^n)$ (Frechet-) Smooth?

Question

I don't know much about Frechet-differentiability. I just need to know if the following statement is true.

Let $p^{-1}+p'^{-1}=1$. Let $g\in L_{p'}(\mathbb{R}^n,\mathbb{R})$. Then the operator

$ G: L_p(\mathbb{R}^n,\mathbb{R})\to \mathbb{R}, \varphi\mapsto \int_{\mathbb{R}^n}\varphi(x)g(x)dx$

is infinitely many times Frechet differentiable.

Any literature regarding problems like these are helpful. Thank you.