The vector space polar 
of a neighborhood 
of 
in a topological vector
space 
is the set
 |
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 
denotes the continuous dual
space of 
(i.e., 
is the space of all continuous linear functionals from 
to the underlying scalar field
of 
).
When 
is a normed space and 
is its unit ball, 
is the closed dual unit
ball. More generally, the Banach-Alaoglu
theorem states that 
is compact in the weak-*
topology for every neighborhood 
of 
.
