We define the $L_p$ norm of a function $f \in B$ (where $B$ is a Banach space) as $||f||_p = (\int |f(x)|^p dx)^{\frac{1}{p}}$. I'm trying to understand if the functional $\phi_p(f) = ||f||_p^2$ is smooth and satisfies the condition of $\alpha$-smoothness wrt to $||.||_B$ i.e, for all $g, g' \in B$ $$\phi_p(g) \le \phi_p(g') + (g-g', \delta_{g'} \phi_p)_B + \frac{\alpha}{2}||g-g'||_B^2$$
What are the structures required on $B$ for this to hold? Assume that $B$ is a family of real-valued functions defined on a space $X \subset R^n$. For the smoothness, let's exclude the zero function.