= Weak boundedness implies norm boundedness
A subset of a normed space is norm bounded if it is bounded under every continuous linear functional. Apply the <Uniform boundedness principle> on the complete dual space to the evaluation family $f\mapsto f(x)$. <Hahn-Banach theorem> identifies the evaluation norm with $\|x\|$, even when the original normed space is not complete. As a consequence, two norms with exactly the same continuous dual as sets must be equivalent: apply this criterion to either norm's unit ball in the other norm.
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