Auerbach's lemma
# 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 ${e_1,$ ..., $e_n}$ of V such that

    $||e_i||$ = 1 and $||e^i||$ = 1 for i = 1, ..., n,

where ${e^1,$ ..., $e^n}$ is a basis of V* dual to ${e_1,$ ..., $e_n},$ i. e. $e^i(e_j)$ = $δ_{ij}.$

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

## 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 ${e_1,$ ..., $e_n}$ be an Auerbach basis of V and ${e^1,$ ..., $e^n}$ corresponding dual basis. By Hahn–Banach theorem each $e^i$ 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).

