Auerbach's lemma explained

In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed vector spaces.

Statement

Let (V, ||·||) be an n-dimensional normed vector space. Then there exists a basis of V such that

||e_i|| = 1 and ||eⁱ|| = 1 for i = 1, ..., n,where is a basis of V* dual to, i. e. eⁱ(e_j) = δ_ij.

A basis with this property is called an Auerbach basis.

If V is an inner product space (or even infinite-dimensional Hilbert space) then this result is obvious as one may take for any orthonormal basis of V (the dual basis is then).

Geometric formulation

An equivalent statement is the following: any centrally symmetric convex body in

R ^{n} has a linear image which contains the unit cross-polytope (the unit ball for the \ell ^{n} _{1} norm) and is contained in the unit cube (the unit ball for the \ell ^{n} _{inf ty} norm).

Corollary

The lemma has a corollary with implications to approximation theory.

Let V be an n-dimensional subspace of a normed vector space (X, ||·||). Then there exists a projection P of X onto V such that ||P|| ≤ n.

Proof

Let be an Auerbach basis of V and corresponding dual basis. By Hahn–Banach theorem each eⁱ extends to f ⁱ ∈ X* such that

||f ⁱ|| = 1.Now set

P(x) = Σ f ⁱ(x) e_i.It's easy to check that P is indeed a projection onto V and that ||P|| ≤ n (this follows from triangle inequality).

References

• Joseph Diestel, Hans Jarchow, Andrew Tonge, Absolutely Summing Operators, p. 146.
• Joram Lindenstrauss, Lior Tzafriri, Classical Banach Spaces I and II: Sequence Spaces; Function Spaces, Springer 1996,, p. 16.
• Reinhold Meise, Dietmar Vogt, Einführung in die Funktionalanalysis, Vieweg, Braunschweig 1992, .
• Przemysław Wojtaszczyk, Banach spaces for analysts. Cambridge Studies in Advancod Mathematics, Cambridge University Press, vol. 25, 1991, p. 75.