Hahn–Banach theorem

The Hahn–Banach theorem extends bounded linear functionals from a subspace to an entire normed vector space without increasing their norm. Its real and complex forms underpin major results in functional analysis.

Hahn–Banach extension theorem: A form of Hahn–Banach that extends a linear functional from a subspace while preserving a bound imposed by a sublinear function. This is the sublinear formulation from which the norm-preserving extension principle follows.
Normed vector space: A vector space equipped with a norm that measures the size of each vector. The theorem's norm-preserving form applies to functionals on these spaces.
Continuous dual space: The vector space of all bounded linear functionals on a normed space. Hahn–Banach guarantees extensions that keep functionals inside this dual space.
Hans Hahn: An Austrian mathematician whose work included an early form of the functional-extension theorem. His 1927 result established an important predecessor to the general theorem.
Open mapping theorem: A theorem stating that a surjective bounded linear map between Banach spaces is open. It is another central Banach-space theorem, but concerns maps between spaces rather than extending functionals.
Sublinear function: A function that is positively homogeneous and subadditive. The real extension theorem controls the functional using a sublinear majorant.
Linear functional: A linear map from a vector space to its scalar field. This is the object extended from the subspace to the whole space.
Bidual space: The continuous dual of the continuous dual of a normed space. Functionals from Hahn–Banach establish the canonical embedding of a space into its bidual.
Stefan Banach: A Polish mathematician who helped found modern functional analysis. His work supplied the setting and influential formulation associated with the theorem.
Uniform boundedness principle: A theorem that turns pointwise boundedness of a family of operators into a uniform norm bound under completeness conditions. It complements Hahn–Banach among the foundational principles of functional analysis.

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