[Math] Are all normed linear spaces Hausdorff? What about a bounded subset of a normed linear space

Question

The proof that the dual space of a normed linear space is complete in proposition 5.4 of chapter on Banach spaces (John Conway, functional analysis) consists of restricting the functionals in the dual space to a bounded subset. The subset is claimed to be Hausdorff. The rest of the proof then follows from isometry of the restriction map and the claim that the space of bounded continuous functions in a Hausdorff space is complete.

If the normed linear space itself is Haudorff, the restriction seems unnecessary.

Answer 1

All normed linear spaces are metric spaces, and metric spaces are Hausdorff.