Corollary 4

Let I be a set and let $(H_i)_{i\in I}$ be a family of closed subgroups of G. The orthogonal of the closed subgroup generated by the $H_i$ is $\bigcap_{i\in I}H^{\bot}_i$. The orthogonal of $\bigcap_iH_i$ is the closed subgroup generated by the subgroups $H^{\bot}_i$.

The first assertion follows from the definition of the orthogonal. Applying this result and Corollary 3 to the family of closed subgroups $(H^{\bot}_i)_{i\in I}$ of $\widehat{G}$, we see that $\bigcap_iH_i$ is the orthogonal of the closed subgroup generated by the subgroups $H^{\bot}_i$, and the second assertion is then obtained by duality.