A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
Since the Banach–Alaoglu theorem is proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC.
In the case of a normed vector space, the polar of a neighbourhood is closed and norm-bounded in the dual space. For example the polar of the unit ball is the closed unit ball in the dual. Consequently, for normed vector space (and hence Banach spaces) the Bourbaki–Alaoglu theorem is equivalent to the Banach–Alaoglu theorem.
For any x in X, let
Since each Dx is a compact subset of the complex plane, D is also compact in the product topology by Tychonoff theorem.
We can identify the closed unit ball in X*, B1(X*), as a subset of D in a natural way:
The claim will be proved if the range of the above map is closed. But this is also clear. If one has a net
in D, then the functional defined by
lies in B1(X*).
If X is a reflexive Banach space, then every bounded sequence in X has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X; or, more succinctly, by applying the Eberlein–Šmulian theorem.) For example, suppose that X=Lp(μ), 1<p<∞. Let fn be a bounded sequence of functions in X. Then there exists a subsequence fnk and an f ∈ X such that
for all g ∈ Lq(μ) = X* (where 1/p+1/q=1). The corresponding result for p=1 is not true, as L1(μ) is not reflexive.