Schauder basis convergence and biorthogonal functionals

Question

Suppose $X$ is a Banach space and has a Schauder basis $(e_{j})_{j=1}^{\infty}$. The associated biorthogonal functionals $(e_{j}^{*})_{j=1}^{\infty}$ are all known to be continuous linear maps and it therefore follows that if $x_{k}\to x\in X$, then \begin{equation} \lim_{k\to\infty}e_{j}^{*}(x_{k})=e_{j}^{*}(x_{0}) \end{equation} for each positive integer $j$. I am wondering if the convergence of the coordinate functionals is not only necessary, but also a sufficient condition for the convergence of $x_{k}$ to $x_{0}$. Namely, if $e_{j}^{*}(x_{k})\to e_{j}^{*}(x_{0})$ for each positive integer $j$, then is it true that $x_{k}\to x_{0}$? This seems perhaps like a standard exercise but I am having some trouble proving it.

Answer 1

This is not even true in a seperable Hilbert space. In this special case, let $\{e_j\}_{j=1}^{\infty}$ be an orthonormal basis so that the corresponding biortogonal maps are $\{\langle \cdot, e_j\rangle\}_{j=1}^\infty$. Then the sequence $\{e_n\}_{n=1}^{\infty}$ clearly does not converge in norm, but for every $j$, $\langle e_n, e_j\rangle \rightarrow 0$ as $n\rightarrow \infty$.