Editing Hahn–Banach theorem (section)

==Converse==
Let {{mvar|X}} be a topological vector space.  A vector subspace {{mvar|M}} of {{mvar|X}} has '''the extension property''' if any continuous linear functional on {{mvar|M}} can be extended to a continuous linear functional on {{mvar|X}}, and we say that {{mvar|X}} has the '''Hahn–Banach extension property''' ('''HBEP''') if every vector subspace of {{mvar|X}} has the extension property.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.  For complete [[metrizable TVS|metrizable topological vector space]]s there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex.{{sfn|Narici|Beckenstein|2011|pp=225-273}}  On the other hand, a vector space {{mvar|X}} of uncountable dimension, endowed with the [[finest vector topology]], then this is a topological vector spaces with the Hahn–Banach extension property that is neither locally convex nor metrizable.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

A vector subspace {{mvar|M}} of a TVS {{mvar|X}} has '''the separation property''' if for every element of {{mvar|X}} such that <math>x \not\in M,</math> there exists a continuous linear functional <math>f</math> on {{mvar|X}} such that <math>f(x) \neq 0</math> and <math>f(m) = 0</math> for all <math>m \in M.</math>  Clearly, the continuous dual space of a TVS {{mvar|X}} separates points on {{mvar|X}} if and only if <math>\{0\},</math> has the separation property.  In 1992, Kakol proved that any infinite dimensional vector space {{mvar|X}}, there exist TVS-topologies on {{mvar|X}} that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on {{mvar|X}}.  However, if {{mvar|X}} is a TVS then {{em|every}} vector subspace of {{mvar|X}} has the extension property if and only if {{em|every}} vector subspace of {{mvar|X}} has the separation property.{{sfn|Narici|Beckenstein|2011|pp=225-273}}