Let $C([a,b],\mathbb{R})$ denote the space of continuous functions from $[a,b]$ to the real numbers. For a function $f\in C([a,b],\mathbb{R})$ and $d\gt 0$, define

$$p_d(f) :=sup\{\lvert f(x)-f(y)\rvert : \lvert x-y\rvert =d,\ x,y \in [a,b]\}.$$

Does this define a family of seminorms on $C([a,b],\mathbb{R})$ indexed by $d$?

If yes, does it follow that $C([a,b],\mathbb{R})$ is locally convex? Which topology does this family induce on $C([a,b],\mathbb{R})$ (if any of the standard topologies)? Thanks.