# Discrete sets in normed spaces

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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What is a "norm discrete set"? Google search turns up your post and nothing else. –  Matt N. Oct 16 '12 at 8:08
@MattN.: Discrete in the norm topology. –  Brian M. Scott Oct 16 '12 at 8:15
@BrianM.Scott Thank you. That of course makes sense. Perhaps one could call it obvious, even. –  Matt N. Oct 16 '12 at 8:19

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.