Every bounded linear functional on a linear subspace of a real normed space extends to the whole space without increasing its norm.
If is a closed linear subspace of a Hausdorff locally convex space and , choose a continuous seminorm with for all , using a basic neighborhood disjoint from . The linear functional is bounded by . The Hahn-Banach theorem extends it to a continuous linear functional with the displayed values. This is the linear-subspace form of separation, sharper than merely distinguishing the two sets.
A real linear functional on a vector subspace, dominated by a sublinear function , can be extended across one new vector by setting . The displayed lower and upper bounds are compatible because . A value between them preserves domination for both signs of . Chain unions and Zorn's lemma then yield the full Hahn-Banach theorem.
A real separable Banach space has a sequence in its dual unit ball with for every . Choose a dense sequence in the unit sphere and apply the Hahn-Banach theorem to obtain . In the complex case use .
For a normed space , the map defined byis a linear isometry. The easy inequality is ; Hahn--Banach extends the norm-one functional on that takes to , proving the reverse inequality.
Point evaluation at zero is bounded on with the essential-supremum norm, and Hahn--Banach extends it to a bounded functional on . No represents this extension: continuous functions supported in shrinking neighbourhoods of zero have value one at zero while their integrals against tend to zero. Hence is strictly larger than .
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The Hahn-Banach theorem is a fundamental result in functional analysis, particularly in the study of linear functionals on normed vector spaces. It has several formulations and applications, but its primary statement concerns the extension of linear functionals. ### Statement of the Hahn-Banach Theorem Informally, the theorem asserts that under certain conditions, a bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space without increasing its norm.