Let $\mathscr F\in\beta\mathbb N$, where $\beta\mathbb N$ is the set on ultrafilters on $\mathbb N$, and $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ the functional which assigns to a bounded sequence $\{a_n\}_{n\in\mathbb N}\subset\mathbb C$, its limit with respect to the ultrafilter $\mathscr F$. Set $$ X=\mathrm{span}\{l_{\mathscr F}: \mathscr F\in\beta\mathbb N\}. $$ It is relatively straightforward to show that $X$ is dense in $\big(\ell^\infty(\mathbb N)\big)^{\!*}$, in the weak-$*$ topology.

My question is whether $X$ is also dense in the dual-norm topology.

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns to a bounded sequence $\{a_n\}_{n\in\mathbb N}\subset\mathbb C$ its limit, with respect to the ultrafilter $\mathscr F$. Set $$ X=\mathrm{span}\,\big\{l_{\mathscr F}: \mathscr F\in\beta\mathbb N\big\}. $$ It is relatively straightforward to show that $X$ is dense in $\big(\ell^\infty(\mathbb N)\big)^{\!*}$, in the weak-$*$ topology.

My question is whether $X$ is also dense in the dual-norm topology.

Let $\mathscr F\in\beta\mathbb N$, where $\beta\mathbb N$ is the set on ultrafilters on $\mathbb N$, and $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ the functional which assigns to a bounded sequence $\{a_n\}_{n\in\mathbb N}\subset\mathbb C$, its limit with respect to the ultrafilter $\mathscr F$. Set $$ X=\mathrm{span}\{l_{\mathscr F}: \mathscr F\in\beta\mathbb N\}. $$ It's relatively straight-forward to show that $X$ is dense in $\big(\ell^\infty(\mathbb N)\big)^{\!*}$, in the weak$^*$ topology.

My question is whether $X$ is also dense in the norm topology.

Let $\mathscr F\in\beta\mathbb N$, where $\beta\mathbb N$ is the set on ultrafilters on $\mathbb N$, and $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ the functional which assigns to a bounded sequence $\{a_n\}_{n\in\mathbb N}\subset\mathbb C$, its limit with respect to the ultrafilter $\mathscr F$. Set $$ X=\mathrm{span}\{l_{\mathscr F}: \mathscr F\in\beta\mathbb N\}. $$ It is relatively straightforward to show that $X$ is dense in $\big(\ell^\infty(\mathbb N)\big)^{\!*}$, in the weak-$*$ topology.

My question is whether $X$ is also dense in the dual-norm topology.