Hahn–Banach theorem

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In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows one to extend bounded linear operators 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. It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the 1920s.

The most general formulation of the theorem needs some preparations. If V is a vector space over the scalar field K (which is either the real numbers R or the complex numbers C), we call a function N : VR 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 the following:

Let N : VR be sublinear, let U be a subspace of V and let φ : UK be a linear functional such that |φ(x)| ≤ N(x) for all x in U. Then there exists a linear map ψ : VK 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: in the case of an infinite dimensional space V, it depends on Zorn's lemma.

In fact, the sublinearity condition on N can be slightly relaxed: it suffices to assume that

N(ax + by) ≤ |a| N(x) + |b| N(y)

for all a and b in K with |a| + |b| = 1 (Reed and Simon, 1980).

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 : UK is continuous and linear, then there exists an extension ψ : VK 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 : VK 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.

References

Lawrence Narici and Edward Beckenstein, 'The Hahn–Banach Theorem: The Life and Times', Topology and its Applications, Volume 77, Issue 2 (3 June 1997) Pages 193-211. An on-line preprint is available here

Michael Reed and Barry Simon, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-5.