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 
.
