The Goldstine-Weston theorem in random normed modules

This article generalize the classical Goldstine-Weston theorem on normed spaces to one on random normed modules: the image of a random normed module $(E,\mathrm{\parallel }\cdot \mathrm{\parallel })$ under the random natural embedding $J$ is dense in its double random conjugate space $E^{\ast \ast }$ with respect to the $(ϵ,\lambda )$ weak star topology; and $J(E)$ is also dense in $E^{\ast \ast }$ with respect to the locally $L^0$-convex weak star topology if $E$ has the countable concatenation property.