According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result - maybe the most important fact about the weak-* topology - [that] echos throughout functional analysis." (Source: Wikipedia)
It is a well-known fact (by Riesz) that the compactness of the unit ball with respect to the norm topology characterizes finite dimensional vector spaces. In a infinite dimensional setting, Banach-Alaoglu recovers the compactness of the unit ball in the weak*-topology which seems to come to relief of a lot of analyst. I have come across a thread recently about the importance of Hahn-Banach which I found very illuminating. I would be interested about the different takes people have on Banach-Alaoglu. Why is it so important? What if we were not to recover the compactness of the unit ball?