In mathematics, particularly in set theory and topology, a "polar set" typically refers to a set that is "small" in some sense, often in relation to a particular topology or concept in analysis. The most common usage of the term "polar set" arises in the context of functional analysis and measure theory.
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For a real dual pair and , the polar set isThe same absolute-value definition applies to a complex pairing. It collects the linear measurements uniformly bounded by one on . Reversing the roles of defines the bipolar.
A polar is convex, balanced and closed in the topology of pointwise convergence on , since it is an intersection of inverse images of closed scalar unit disks under linear evaluations. Inclusion reverses: implies .
In with the usual dot pairing, the polar of is , because the maximum of on that square is . In a normed space paired with its continuous dual, the polar of the primal closed unit ball is exactly the dual closed unit ball by the definition of the operator norm.