Question

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.

Answer 1

The $\rho_d$ are indeed seminorms; this can be verified by direct calculations or by noting that

1. $\sup\lbrace |g(x,y)|:|x-y|=d, x,y\in[a,b]\rbrace$ defines a seminorm on $C([a,b]^2,\mathbb{R})$ (as a supremum of function values on a set), and

2. the map $F$, defined by $F(f)(x,y)=f(x)-f(y)$, is linear; from $C([a,b],\mathbb{R})$ to $C([a,b]^2,\mathbb{R})$.

Every topology induced by seminorms is locally convex because the seminnorm-balls are convex.

Finally, every constant function gets seminorm zero for all $d$, so this topology is not Hausdorff and therefore not equal to any standard topology on $C([a,b],\mathbb{R})$ that I am aware of.