We consider the problem of superreplication under Knightian uncertainty in a discrete‐time financial market. In the absence of a reference probability measure, we develop a duality theory which is based on a locally convex vector space and allows to treat measurable quantities without further topological restrictions. We obtain the existence of an optimal strategy and a duality relation between (non‐equivalent) martingale measures and superreplicable claims. The continuum hypothesis plays an important role in our approach.