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If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ are discontinuous?

More generally (or is it equivalent?) does there exists a Hamel basis with the property that for all finite sets $N\subset\Lambda$, the closed span of $(x_\alpha)_{\alpha\in\Lambda\setminus N}$ is dense in $X$?

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    $\begingroup$ Take a base of open sets for the topology of minimal cardinality. Show that there is a linearly independent set that intersects each basic open set in an infinite set. You can do this easily by transfinite induction. Any extension of that linearly independent set to a Hamel basis has the properties you want; in fact, if you throw away any finite subset of the Hamel basis, the remaining vectors will be dense in the space. $\endgroup$
    – Bill Johnson
    Commented May 20, 2021 at 17:23
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    $\begingroup$ To do the construction, the only fact you need is that the dimension of the space is at least as large as the density character of the space, so the space can be any infinite dimensional normed space. $\endgroup$
    – Bill Johnson
    Commented May 20, 2021 at 17:23

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