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.
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 (or even infinite-dimensional Hilbert space) then this result is obvious as one may take for any orthonormal basis of (the dual basis is then ).
An equivalent statement is the following: any centrally symmetric convex body in has a linear image which contains the unit cross-polytope (the unit ball for the norm) and is contained in the unit cube (the unit ball for the norm).
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 HahnâÂÂBanach 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.
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 .
Let be an Auerbach basis of and corresponding dual basis. By the HahnâÂÂBanach theorem each extends to such that . Now set . It is easy to check that is indeed a projection onto and that (this follows from the triangle inequality).