Let $\mathcal A'$ and $\mathcal A''$ be the dual and bidual spaces of a locally convex algebra $\mathcal A$ with dual and weak${}^*$ topology, respectively. In this paper, we show that $\mathcal A$ has a bounded right (left) approximate identity if and only if $\mathcal A''$ has a right (left) unit with respect to the first (second) Arens product.