Summary: Recently a new hyperspace topology with good applications in minimization problems was introduced. This topology is called bounded proximal topology [G. Beer, R. Lucchetti, Trans. Am. Math. Soc. 335, No. 2, 805-822 (1993; Zbl 0810.54011)]. We study properties of the bounded proximal topology and the upper Attouch-Wets topology applied on function spaces. These topologies exhibit a good behaviour on a continuous dual \(X^*\) of a linear metrizable space \(X\).