Is the Whitney topology finer or coincide with uniform topology for codomain is $\mathbb{R}$

Question

Let $C^{r}(U,\mathbb{R})$ be the space of $C^{r}$ functions from open subset $U\subseteq\mathbb{R}^{n}$ to $\mathbb{R}$, $0\leq r<\infty$. This space can equipped with the Whitney topology(or strong topology), is this topology finer than the uniform topology(supremum norm as a metric), or actually coincide?

Edit It seems that there is no way to compare this two topology in the above setting. If we only focus on subspace $C^{r}_{b}(U,\mathbb{R})$, the space of $C^{r}$ and bounded function, so that supremum norm can be defined. Can we compare their topology?

Answer 1

This question is discussed in detail in section 1.7 of the book:

Eldering, Jaap. Normally hyperbolic invariant manifolds: the noncompact case. Vol. 2. Atlantis Press, 2013.

Here is a preprint, and here is the final paragraph of section 1.7:

We conclude that the $C^k_b$-topologies induced by our uniform norms are not equivalent to either the weak or strong Whitney topology, because the weak topology allows arbitrary behavior of functions outside compact sets, while the strong topology completely restricts that behavior. Our norms allow moderate variations at infinity. In general, ‘moderate behavior’ is not well-defined on a general noncompact manifold, as it depends on the choice of charts. In the setting of bounded geometry, though, the uniform, metric structure makes this behavior unambiguous; we can restrict to normal coordinate charts and consider ‘moderate behavior’ with respect to these. Note that these topologies are equivalent on compact domains.