Let $D$ be a discrete set in the norm topology of a normed space $X$. Does $D$ remain discrete in the weak topology of $X$? What if $D$ is moreover closed?
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No. For an example, consider $X=\ell_2(\mathbb N)$ and $D=\lbrace 0, e_1,e_2,\ldots \rbrace$ with the usual unit vectors. Then $D$ is norm-discrete as the distance between two points in $D$ is at least $1$ but $D$ is not weakly dicrete because $e_n \to 0$ weakly.