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 $\Vert x\Vert$, 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.