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When trying to prove that a linear functional is bounded iff it is Lipschitz continuous, is it true that if we assume that a linear functional is Lipschitz continuous on a normed linear space $X$, then it is also continuous on $X$?

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    @Davide Hmm… It appears I did. Oops!2013-03-11

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In view of t.b.'s comment, the answer is concluded to be yes. In essence, a sequential continuity argument seems sufficient.