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, ..., e_n} of V such that

||e_i|| = 1 and ||eⁱ|| = 1 for i = 1, ..., n,

where {e¹, ..., eⁿ} is a basis of V* dual to {e₁, ..., e_n}, 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 {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 {e₁, ..., e_n} 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) 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.

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