TITLE: Dual space of the intersection of locally convex vector spaces
QUESTION [2 upvotes]: 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.

REPLY [0 votes]: $\DeclareMathOperator\span{span}$Let $M$ be a set of finite measures on $(X,\mathcal{A})$ and consider
$$
L_p(X,\mathcal{A},M) := \left(\bigcap_{\mu \in M} \mathcal{L}_p(X,\mathcal{A},\mu) \right) \, / \, [M]
$$
for some $p\in\mathopen]1,\infty\mathclose[.$ Let $q\in\mathopen]1,\infty\mathclose[$ such that $1/p + 1/q = 1$.
Then, the dual space of $(L_p(X,\mathcal{A},M),\tau(\|\cdot\|_{\mu,p} : \mu \in M))$ is given by
$$
L_p(X,\mathcal{A},M)^* = \span \left( \bigcup_{\mu\in M} \bigl\{ (\cdot,g)_\mu|_{L_p(X,\mathcal{A},M)} : g \in \mathcal{L}_q(X,\mathcal{A},\mu)\, / \, [M]\bigr\} \right).
$$
Proof (similar to Satz II.2.4 in Werner (2018) "Funktionalanalysis", Springer-Verlag): Clearly, for each $\mu \in M$ and each $g \in \mathcal{L}_q(X,\mathcal{A},\mu)$ we have that $(\cdot,g)_\mu|_{L_p(X,\mathcal{A},M)}$ is a continuous linear function on $L_p(X,\mathcal{A},M)$.
Conversely, let $f^* \in L_p(X,\mathcal{A},M)^*$ and define
$$
\nu(A) := f^*(\mathbb{1}_A) \quad A \in\mathcal{A}.
$$
Then, $\nu$ is a signed measure. Since $f^*$ is  $\tau(\|\cdot\|_{\mu,p} : \mu \in M)$-continuous there are a finite set $F \subset M$ and a positive $L > 0$ such that
$$
|f^*(g)| \leq L \max \{ \|g\|_{\mu,p} : \mu \in F\}
$$
for all $g \in L_p(X,\mathcal{A},M),$ which yields that $\nu \ll \sum_{\mu \in F} \mu$. According to the Radon–Nikodym theorem there is a density $h$ of $\nu$ w.r.t. the finite measure $\sum_{\mu \in F} \mu$, that is,
$$
f^*(g) = \int g h \, \mathrm{d} (\textstyle{\sum_{\mu \in F} \mu})
$$
for all $g \in L_{\infty}(\sum_{\mu \in F} \mu)$. Then, one can show that $h \in \mathcal{L}_{q}$ (see for example proof of Satz VIII.2.3 in Werner (2018)) and $f^* = \sum_{\mu \in F} (\cdot,h)_\mu|_{L_p(X,\mathcal{A},M)}$ on $L_p(X,\mathcal{A},M)$ since $L_{\infty}(\sum_{\mu \in F} \mu)$ is dense in $L_{p}(\sum_{\mu \in F} \mu)$.
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