Strong topology of $C_{\lambda}(X)$

Let $X$ be a completely regular space. A subset $A$ of $X$ is called bounded if the number set $f(A)$ is bounded for each continuous function $f$ on $X$. Let $\lambda$ be some family of bounded subsets of $X$. By definition, $C_{\lambda}(X)$ is the space of all real-valued continuous functions on $X$, its topology being the topology of uniform convergence on each set of $\lambda$. It is proved that the strong topology (in the sense of the theory of topological vector spaces) of $C_{\lambda}(X)$ is the topology of bounded convergence on $X$ (i.e. that of uniform convergence on each bounded subset of $X$).