# 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 [math]\displaystyle{ \mathbf{R}^n }[/math] has a linear image which contains the unit cross-polytope (the unit ball for the [math]\displaystyle{ \ell_1^n }[/math] norm) and is contained in the unit cube (the unit ball for the [math]\displaystyle{ \ell_{\infty}^n }[/math] 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).

## 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, ISBN:3540606289, p. 16. - Reinhold Meise, Dietmar Vogt,
*Einführung in die Funktionalanalysis*, Vieweg, Braunschweig 1992, ISBN:3-528-07262-8. - Przemysław Wojtaszczyk,
*Banach spaces for analysts*. Cambridge Studies in Advancod Mathematics, Cambridge University Press, vol. 25, 1991, p. 75.

Original source: https://en.wikipedia.org/wiki/Auerbach's lemma.
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