Given a Banach space $X$.
Denote the annihilator: $$A\subseteq X:\quad A^\perp:=\{x'\in X':x'A=0\}$$
Then does it hold: $$Z_\lambda=\overline{Z_\lambda}\leq X:\quad\overline{\left\langle\bigcup_{\lambda\in\Lambda}Z_\lambda^\perp\right\rangle}=\left(\bigcap_{\lambda\in\Lambda}Z_\lambda\right)^\perp$$ How can I check this?
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Here is a counterexample, consider $C([0,1])$. Note that $C([0,1])^*$ is given by bounded measures on $[0,1]$. Now look at
$$\Lambda=\left\{\text{finite subsets of }[0,1]\right\}\\ Z_\lambda = \{f\in C([0,1]) \mid f\lvert_\lambda = 0\}$$ Then you get the following: $$\bigcap_{\lambda} Z_\lambda = Z_{\,\bigcup \lambda}\\ Z_\lambda^\perp=\left\{\mu\in C([0,1])^*\mid \mu\left(\overline{\lambda}\right)=0\right\}$$ Since $\lambda\in\Lambda$ consists of isolated points $Z_\lambda^\perp$ always consists of linear combinations of dirac measures on $\lambda$. So for example the standard Lebesgue measure $\mu = dx$ does not lie in $\overline{\left\langle\bigcup_{\lambda\in\Lambda}Z_\lambda^\perp\right\rangle}$, but clearly it does in $\left(\bigcap_{\lambda}Z_\lambda\right)^\perp=Z_{[0,1]}^\perp=\{0\}^\perp$.