Example 3

Let F be normed space; the family of functions $\|x-\mathbf{a}\|^\alpha$ ($\alpha$ an arbitrary real number) is a comparison scale when $\mathfrak{F}$ is the trace on the complement of $\mathbf{a}$ of the filter of neighbourhoods of this point. Note that if $p$ and $q$ are two distinct norms on F, the union of the two comparison scales $(p(x-\mathbf{a}))^\alpha$ and $(q(x-\mathbf{a}))^\alpha$ is not in general a comparison scale.