Equality on union of an increasing sequence of subspaces of vector space and second dual Let $V$ be a normed vector space over $\mathbb{C}$ and $V_n$ be an increasing sequence$(V_n \subset V_{n+1})$ of subspaces of $V$. Does the following equality holds
in $V^{**}$

$\left (\overline {\bigcup_n V_n}\right)^{**}= \overline {\bigcup_n V_n^{**}}$

Basically we need to show that $\bigcup_n V_n^{**}$ is dense in $\left (\overline {\bigcup_n V_n}\right)^{**}$. Any ideas?

This is not true. To see this take $V=c_0$ and let $e_1 , e_2 , ....,$ where $e_i= (0,0,...0,1,0...)$. Denote $V_n =\mbox{span}\{e_1, ..., e_n \}.$ It iis easy to see that $$\overline{\bigcup V_n } =c_0 $$ and hence $$\left(\overline{\bigcup V_n } \right)^{**} =\ell^{\infty} .$$
But since $V_n $ are finite dimensional thus $$(V_n , ||\cdot ||_{\infty})^{**} =(V_n , ||\cdot ||_{\infty} )$$ shortly $$V_n^{**} = V_n $$ since on finite dimensional spaces all norms are equivalent. But then $$\overline{\bigcup V_n^{**} } =\overline{\bigcup V_n } \neq \ell^{\infty}$$ since sequence $$(1, 1, ....,1,..)$$
connot be approximated (in the supremum norm) by any sequence with only finite number of coordinates different to zero.
