In functional analysis, the Banach-Alaoglu theorem (also sometimes called Alaoglu's theorem) is a result which states that the
norm unit ball of the continuous
dual 
of a topological vector space 
is compact in the weak-*
topology induced by the norm topology on 
.
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).
The proof for a general topological vector space 
was proved by Alaoglu in the 1940s though the special case
for 
separable was proved by Banach in the 1930s. Since
then, the theorem has been generalized to other miscellaneous contexts, most notably
by Bourbaki into the language of dual topologies, and 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.
