 Weighted metric for local uniform convergence

Let $X$ have nested compact sets $K_n$ covering it, with $K_n\subseteq \mathrm{int}{K}_{n+1}$. Then every point has a neighborhood in some $K_n$, and every compact subset is contained in one $K_n$. For continuous scalar functions, take positive weights with ${\sum }_n{w}_n<\infty$ and defineThe bounded metric transform makes the sum finite and gives its triangle inequality; the covering property gives definiteness. On $\mathbb{R}$, one may use $K_n=[-n,n]$ and $w_n=2^{-n}$.

If $\sigma (f_j,f)\to 0$, each fixed compact supremum norm tends to zero, since its summand is bounded by the whole metric. Conversely, control finitely many compact supremum norms and then the summable weight tail. This proves that $\sigma$ realizes locally uniform convergence under the stated exhaustion conditions.

Merely writing a space as a union of compact sets is insufficient. Enumerate the points of the compact space $\{0\}\cup \{1/n:n\ge 1\}$ and take finite initial subsets as $K_n$. The resulting metric detects pointwise control on those finite sets, not uniform control on the entire compact space. The neighborhood and compact-containment conditions explain the difference.