Geodesic
Strong topology of $C_{\lambda}(X)$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 5 (1998), pp. 67-75

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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 $X$). 
@article{TIMM_1998_5_a4,
     author = {N. V. Velichko},
     title = {Strong topology of $C_{\lambda}(X)$},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {67--75},
     publisher = {mathdoc},
     volume = {5},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_1998_5_a4/}
}
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N. V. Velichko. Strong topology of $C_{\lambda}(X)$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 5 (1998), pp. 67-75. http://geodesic.mathdoc.fr/item/TIMM_1998_5_a4/