Why is it important?

The Hahn–Banach theorem is the foundation of many major results in functional analysis because it allows us to construct enough continuous linear functionals to study normed spaces. It is used to prove:

The existence of continuous linear functionals.

Separation theorems for convex sets.

The duality theory of Banach spaces.

The weak and weak-* topologies.

The Banach–Alaoglu theorem (indirectly).

The Krein–Milman theorem (through separation arguments).
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