Editing Hahn–Banach theorem (section)

===Supporting hyperplanes===

Since points are trivially [[Convex set|convex]], geometric Hahn–Banach implies that functionals can detect the [[Boundary (topology)|boundary]] of a set.  In particular, let <math>X</math> be a real topological vector space and <math>A \subseteq X</math> be convex with <math>\operatorname{Int} A \neq \varnothing.</math> If <math>a_0 \in A \setminus \operatorname{Int} A</math> then there is a functional that is vanishing at <math>a_0,</math> but supported on the interior of <math>A.</math><ref name="Zalinescu" />

Call a normed space <math>X</math> '''smooth''' if at each point <math>x</math> in its unit ball there exists a unique closed hyperplane to the unit ball at <math>x.</math> Köthe showed in 1983 that a normed space is smooth at a point <math>x</math> if and only if the norm is [[Gateaux derivative|Gateaux differentiable]] at that point.{{sfn|Narici|Beckenstein|2011|pp=177-220}}