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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?

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I recall that during my undergraduate degree being told that compactness was important but we'll only realise why later.

It's important in the same way that finiteness is important in arithmetic. We can't add infinite numbers (well, until Cantor came along), but we can add finite numbers.

In the same way, compactness is a replacement for finiteness in the topological context: from any open covering we can extract a finite cover. It's best to see this in action in a major theorem such as integrating over a manifold where we use a partition of unity.

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It allows you to cook up a subsequences almost anywhere and it very often turns out to be nice to have such a subsequence. For example in probability theory, since the set of probability measures(since they have measure equal to 1) it is weak$^*$-compact. So you can always get a convergent subsequence!

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