Topologies β 0 , β 1 , β , β ∞ , β ∞ c {\beta _0},{\beta _1},\beta ,{\beta _\infty },{\beta _{\infty c}} are defined on C b ( X , E ) {C_b}(X,E) , the space of all bounded, continuous functions from a completely regular Hausdorff space X, into E, a normed space, and their duals are determined. Also many properties of these topologies are proved.