Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}_s),$ that is, the vector space of all $\mathcal{T}_s$-continuous linear functionals on $E_s.$ Furthermore, we set $M := \bigcap_{s \in S} E_s.$
The goal is to characterize the dual space of $(M,\mathcal{T}_M)$, where $\mathcal{T}_M := \tau(\mathcal{T}_s \cap M: \,s\in S)$ denotes the smallest topology on $M$ that contains $\mathcal{T}_s \cap M$ for every $s \in S.$
Obviously, we have
\begin{equation} \text{lin span}\, \bigg(\bigcup_{s\in S} E_s^*|_M\bigg) \subset M^*, \end{equation}
where $E_s^*|_M$ denotes the set of all continuous linear functionals of $E_s^*$ restricted to $M.$
Question (general): Do we even have
\begin{equation} \text{lin span}\, \bigg(\bigcup_{s\in S} E_s^*|_M\bigg) = M^*\, ? \end{equation}
Question (specific): Is something known for the dual space of the intersection of Lebesgue spaces like $L_p(X,\mathcal{A},\mu) \cap L_p(X,\mathcal{A},\nu),$ where $\mu$ and $\nu$ are $\sigma$-finite measures?