Corollary 3

The product of a family of approximation spaces is an approximation space.

Let indeed $(E_i)_{i\in I}$ be a family of approximation spaces. For every finite subset J of I, let us put $E_J=\prod_{i\in J}E_i$ and denote by $v_J$ the canonical mapping of $E =\prod_{i\in I}E_i$ into $E_J$. The locally convex space $E_J$ is an approximation space (Remark 1). The corollary then follows from Prop. 12 applied to the locally convex space E, to the family $(E_J)$ of locally convex spaces, and to the continuous linear mappings $v_J: E\rightarrow E_J$.