Auerbach's lemma

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 {e1, ..., en} of V such that
 * ||ei|| = 1 and ||ei|| = 1 for i = 1, ..., n,

where {e1, ..., en} is a basis of V* dual to {e1, ..., en}, i. e. ei(ej) = δij.

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 {ei} any orthonormal basis of V (the dual basis is then {(ei|·)}).

Geometric formulation
An equivalent statement is the following: any centrally symmetric convex body in $$ \mathbf{R}^n $$ has a linear image which contains the unit cross-polytope (the unit ball for the $$\ell_1^n$$ norm) and is contained in the unit cube (the unit ball for the $$\ell_{\infty}^n $$ 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 {e1, ..., en} be an Auerbach basis of V and {e1, ..., en} 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) ei.

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