Example

— Let X be a locally compact space and $E = \mathscr{K}(X; \mathbf{R})$ the vector space of finite continuous real valued functions defined over X with compact support. For every compact subset K of X, let $E_K$ be the vector subspace of E formed by those functions $f \in E$ which are such that $x \notin K \Rightarrow f(x) = 0$. Denote by $\mathscr{T}_K$ the topology induced on $E_K$ and by $\mathscr{T}_u$ the topology of uniform convergence on X. The inductive limit $\mathscr{T}$ of the topologies $\mathscr{T}_K$ is finer than $\mathscr{T}_u$; we can show that if X is paracompact and not compact, then $\mathscr{T}$ is strictly finer than $\mathscr{T}_u$ (cf. INT, III, 2nd ed., § 1.8). The importance of $\mathscr{T}$ lies in the fact that the linear forms on E that are continuous in $\mathscr{T}$ are precisely the real measures on X (INT, III, 2nd., § 1.3).