The Banach-Alaoglu theorem (also sometimes called Alaoglu's theorem) states that the closed unit ball of
the continuous dual 
of a normed vector
space 
is compact in the weak-*
topology 
.
More precisely, given a topological vector space 
and a neighborhood 
of 
in 
, the Banach-Alaoglu theorem says that the so-called polar 
of 
is weak-* compact (i.e.,
is compact in the above-mentioned weak-*
topology of 
)
where
 |
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).
Rudin (1991, p. 401) describes Alaoglu's proof and Banach's earlier result for separable Banach
spaces. Bourbaki (2002, Chaps. 3-4) treats the theorem in the language of
dual topologies. The theorem has a number of significant
corollaries. For example, the theorem implies that
every bounded sequence in
a reflexive Banach
space 
(e.g., when 
is a Hilbert space) has a weakly
convergent subsequence and hence that the norm-closures of bounded convex
sets in such spaces are weakly compact.
Worth noting is that the Banach-Alaoglu theorem has a sort of converse which is also true. In particular, if 
is a Banach space with dual 
, if 
denotes the closed unit
ball in 
,
and if 
is a convex set in 
for which the intersection 
is weak-* compact for every 
, then 
is necessarily weak-* closed.
