A bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space without increasing its norm. Named for Hans Hahn and Stefan Banach, it is one of the pillars of functional analysis, guaranteeing that normed spaces have a rich supply of continuous linear functionals.
Facts
StatementThe Hahn-Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. 1 Proof YearHahn had already proved a norm-preserving version for Banach spaces in 1927 per the same History section; the dominated-extension form now called the Hahn-Banach theorem is Banach's 1929 generalization, the year recorded here. Classification
Statement Form Connections
Sources
1. Hahn-Banach Theorem (Wikipedia)
Wikimedia Foundationlead section, first paragraph
In functional analysis, the Hahn-Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space.
History section, on Banach's 1929 generalization
In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses sublinear functions.
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