Given a topological vector space and a neighborhood of in , the polar of is defined to be the set
and where denotes the magnitude of the scalar in the underlying scalar field of (i.e., the absolute value of if is a real vector space or its complex modulus if is a complex vector space) and where denotes the continuous dual space of (i.e., is the space of all continuous linear functionals from to the underlying scalar field of ).
Worth noting is that the polar is essentially the norm unit ball in and is fundamental in functional analysis, e.g., in the Banach-Alaoglu theorem which says is weak-* compact for all neighborhoods of in .