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 be an -dimensional normed vector space. Then there exists a basis of such that
and for,
where is a basis of dual to, i.e..
A basis with this property is called an Auerbach basis.
If is an inner product space then this result is obvious as one may take for any orthonormal basis of .

Geometric formulation

An equivalent statement is the following: any centrally symmetric convex body in has a linear image which contains the unit cross-polytope and is contained in the unit cube.

Proof

By induction on the dimension. Pick an arbitrary unit vector. Because the set of norm-1 points make up a convex symmetric body in, there exists a hyperplane supporting at. This is a consequence of the hyperplane separation theorem, which is a consequence of the Hahn–Banach theorem.
Now, define the dual vector, such that. That is, the contour surfaces of are parallel to.
Then, the subspace is a normed space of dimension, and apply induction.

Corollary

The lemma has a corollary with implications to approximation theory.
Let be an -dimensional subspace of a normed vector space. Then there exists a projection of onto such that.

Proof

Let be an Auerbach basis of and corresponding dual basis. By the Hahn–Banach theorem each extends to such that
.
Now set
.
It is easy to check that is indeed a projection onto and that .