Hahn–Banach theorem

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In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of 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.

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[edit] Formulation

The most general formulation of the theorem needs some preparation. Given a vector space V over the scalar field \scriptstyle\mathbb{K} (either the real numbers \scriptstyle\mathbb{R} or the complex numbers \scriptstyle\mathbb{C}), a function \scriptstyle\mathcal{N}:V\rightarrow\mathbb{R} is called sublinear if

\mathcal{N}(ax+by)\leq|a|\mathcal{N}(x) + |b|\mathcal{N}(y)\qquad\forall x,y\in V\quad\forall a,b\in\mathbb{K}.

As it can be easily proven, every norm and every seminorm on V is sublinear, however these functions are not the only examples of sublinear functions.

The Hahn–Banach theorem states that if \scriptstyle\mathcal{N}:V\rightarrow\mathbb{R} is a sublinear function, and \scriptstyle\varphi:U\rightarrow\mathbb{K} is a linear functional on a subspace U of V which is dominated by \scriptstyle\mathcal{N} on U i.e.

|\varphi(x)|\leq\mathcal{N}(x)\qquad\forall x \in U

then there exists a linear extension \scriptstyle\psi:V\rightarrow\mathbb{K} of φ to the whole space V i.e. there exists linear functional ψ such that

\psi(x)=\varphi(x)\qquad\forall x\in U

and

|\psi(x)|\leq\mathcal{N}(x)\qquad\forall 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, one formulation of the axiom of choice.

It is possible to relax slightly the sublinearity condition on \scriptstyle\mathcal{N}, requiring only that

\mathcal{N}(ax+by)\leq|a|\mathcal{N}(x) + |b|\mathcal{N}(y)\qquad\forall x,y\in V\quad |a|+|b|=1\in\mathbb{R}

according to (Reed and Simon, 1980). This reveals the intimate connnection between the Hahn–Banach theorem and convexity.

The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.

[edit] Important consequences

The theorem has several important consequences, some of are which 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|| = 1/dist(z,U).

[edit] Hahn-Banach separation theorem

Another version of Hahn-Banach theorem is known as Hahn-Banach separation theorem.[1][2] It has numerous uses in complex geometry.[3]

Theorem: Let V be a topological vector space over \scriptstyle{\Bbb K}={\Bbb R}\text{ or }{\Bbb C}, and A, B convex, non-empty subsets of V. Assume that \scriptstyle A\cap B=\varnothing. Then

(i) If A is open, there exists a continuous linear map \scriptstyle\lambda:\; V\mapsto {\Bbb K} and \scriptstyle t\in {\Bbb R} such that

Re \ \lambda(a) < t \leq Re\ \lambda(b)

for all \scriptstyle a\in A, b \in B

(ii) If V is locally convex, A is compact, and B closed, there exists a continuous linear map \scriptstyle\lambda:\; V\mapsto {\Bbb K} and \scriptstyle s, t\in {\Bbb R} such that

Re \ \lambda(a) < t < s < Re\ \lambda(b)

for all \scriptstyle a\in A, b \in B.

[edit] Relation to the axiom of choice

As mentioned earlier, the axiom of choice implies the Hahn–Banach theorem. The converse is not true. One way to see that is by noting that the ultrafilter lemma, which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case. The Hahn–Banach theorem can in fact be proved using even weaker hypotheses than the ultrafilter lemma.[4] For separable Banach spaces, Brown and Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic.[5]

[edit] See also

[edit] References

  • Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.

[edit] Notes

  1. ^ Klaus Thomsen, The Hahn-Banach separation theorem, Aarhus University, Advanced Analysis lecture notes
  2. ^ Gabriel Nagy, Real Analysis lecture notes
  3. ^ R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.
  4. ^ D. Pincus, The strength of Hahn–Banach's Theorem, in: Victoria Symposium on Non-standard Analysis, Lecture notes in Math. 369, Springer 1974, pp. 203-248. Citation from M. Foreman and F. Wehrung, The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, "Fundamenta Mathematicae" 138 (1991), p. 13-19.
  5. ^ D. K. Brown and S. G. Simpson, Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Annals of Pure and Applied Logic, 31, 1986, pp. 123-144. Source of citation.