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 w.r.t. $\sigma$-finite measures? For example, let $\mu$ and $\nu$ be $\sigma$-finite measures, what is known about the dual of the quotient space $$ L_p(X,\mathcal{A},\{\mu,\nu\}) := \left(\mathcal{L}_p(X,\mathcal{A},\nu) \cap \mathcal{L}_p(X,\mathcal{A},\mu)\right) \, /\,\, [\mu+\nu], $$ equipped with the topology $\tau(\|\cdot\|_{\mu,p},\|\cdot\|_{\nu,p})?$ Here, $[\mu+\nu]$ is the subspace of all functions $f$ such that $f = 0$ almost surely w.r.t. $\mu+\nu$ and $\mathcal{L}_p(X,\mathcal{A},\nu)$ denotes the set of all $\mathcal{A}$-measurable functions that are $p\text{th}$ power $\nu$-integrable.