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Hahn-Banach theorem

The Hahn-Banach theorem is a central tool in functional analysis; it shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space interesting.

The most general formulation of the theorem needs some preparations. If V is a vector space over the scalar field K (either the real numbers R or the complex numbers C), we call a function N : V -> R sublinear if N(ax + by) ≤ |a| N(x) + |b| N(y) for all x and y in V and all scalars a and b in K. Every norm on V is sublinear, but there are other examples.

The Hahn-Banach theorem states that:

Let N : V -> R be sublinear, let U be a subspace of V and let φ : U -> K be a linear functional such that |φ(x)| ≤ N(x) for all x in U. Then there exists a linear map ψ : V -> K which extends φ (meaning ψ(x) = φ(x) for all x in U) and which is dominated by N on all of V (meaning |ψ(x)| ≤ N(x) for all x in V).

The extension ψ is in general not uniquely specified by φ and the proof gives no method as to how to find ψ: it depends on Zorn's lemma.

Several important consequences of the theorem are also sometimes called "Hahn-Banach theorem":

  • If V is a normed vector space with subspace U (not necessarily closed) and if φ : U -> K is continuous and linear, then there exists an extension ψ : V -> K of φ which is also continuous and linear and which has the same norm as φ (see Banach space for a discussion of the norm of a linear map).
  • If V is a normed vector space with subspace U (not necessarily closed) and if z is an element of V not in the closure of U, then there exists a continuous linear map ψ : V -> K with ψ(x) = 0 for all x in U, ψ(z) = 1, and ||ψ|| = ||z||-1.

The Mizar project has completely formalized and automatically checked the proof of the Hahn-Banach theorem in the HAHNBAN file (http://mizar.uwb.edu.pl/JFM/Vol5/hahnban).

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