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Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".

Hahn–Banach theorem: — If p: V → R {\displaystyle p:V\to \mathbb {R} } is a sublinear function, and φ: U → R {\displaystyle \varphi:U\to \mathbb {R} } is a linear functional on a linear subspace U ⊆ V {\displaystyle U\subseteq V} which is dominated by p {\displaystyle p} on U {\displaystyle U}; that is, φ (x) ≤ p (x) ∀ x ∈ U {\displaystyle \varphi (x)\leq p(x)\qquad \forall x\in U} then there exists a linear extension ψ: V → R {\displaystyle \psi:V\to \mathbb {R} } of φ {\displaystyle \varphi } to the whole space V {\displaystyle V} which is dominated by p {\displaystyle p} on V {\displaystyle V}; that is, there exists a linear functional ψ {\displaystyle \psi } such that ψ (x) = φ (x) ∀ x ∈ U, ψ (x) ≤ p (x) ∀ x ∈ V. {\displaystyle {\begin{aligned}\psi (x)&=\varphi (x)&\forall x\in U,\\\psi (x)&\leq p(x)&\forall x\in V.\end{aligned}}}