w*-convergence vs. convergence on a dense subspace

Question

Let us have a Banach space $X$, a dense subspace $D\subseteq X$, a net $\{\phi_{i}\colon i\in\mathcal I\}$ in $X^*$ and $\phi\in X^*$. Suppose that $$\lim\limits_{i\in\mathcal I}\phi_{i}(d)=\phi(d)$$ for every $d\in D$. Does it follow that $$\lim\limits_{i\in\mathcal I}\phi_{i}(x)=\phi(x)$$ for every $x\in X$?

Answer 1

If $\sup_{i\in I}\lVert \phi_i\rVert_{X^*}$ is finite, it is true.

If $\lim_{i\in I}\phi_i(x)=\phi(x)$ then by the uniform boundedness principle, $\sup_{i\in I}\lVert \phi_i\rVert_{X^*}$ is finite.

There are cases where $\lim_{i\in I}\phi_i(d)=\phi(d)$ for every $d\in D$, $D$ dense in $X$ but $\sup_{i\in I}\lVert \phi_i\rVert_{X^*}$ is not finite.

For example, take $X=\ell^1$ endowed with the usual norm, $D$ the set of the sequences with finite support and $\phi_j(x)=j^2x_j$. For $x\in D$, $\phi_j(x)=0$ if $j$ is large enough and if $x$ is such that $x_j=\frac 1{j^2}$, then $\varphi_j(x)=1$ for any $j$.