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.
If V is an inner product space (or even infinite-dimensional Hilbert space) then this result is obvious as one may take for {ei} any orthonormal basis of V (the dual basis is then {(ei|·)}).
An equivalent statement is the following: any centrally symmetric convex body in
Let V be an n-dimensional subspace of a normed vector space (X, ||·||).
By Hahn–Banach theorem each ei extends to f i ∈ X* such that Now set It's easy to check that P is indeed a projection onto V and that ||P|| ≤ n (this follows from triangle inequality).