Hahn-Banach theorem

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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 ( though that is a matter of opinion ).

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.

Now let U be a subspace of V and let φ : U -> K be a linear function such that |φ(x)| ≤ N(x) for all x in U. Then the Hahn-Banach theorem states that 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 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 x0 is an element of V not in U, then there exists a continuous linear map ψ : V -> K with ψ(x) = 0 for all x in U, ψ(x0) = 1, and ||ψ|| = ||x0||-1.