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.

Let ${\displaystyle (V,\|\cdot \|)}$ be an ${\displaystyle n}$-dimensional normed vector space. Then there exists a basis ${\displaystyle \{e_{1},\dots ,e_{n}\}}$ of ${\displaystyle V}$ such that ${\displaystyle \|e_{i}\|=1}$ and ${\displaystyle \|e^{i}\|=1}$ for ${\displaystyle i=1,\dots ,n}$, where ${\displaystyle \{e^{1},\dots ,e^{n}\}}$ is a basis of ${\displaystyle V^{*}}$ dual to ${\displaystyle \{e_{1},\dots ,e_{n}\}}$, i.e. ${\displaystyle e^{i}(e_{j})=\delta _{ij}}$.

A basis with this property is called an Auerbach basis.

If ${\displaystyle V}$ is an inner product space (or even infinite-dimensional Hilbert space) then this result is obvious as one may take for ${\displaystyle \{e_{i}\}}$ any orthonormal basis of ${\displaystyle V}$ (the dual basis is then ${\displaystyle \{(e_{i}|\cdot )\}}$).

An equivalent statement is the following: any centrally symmetric convex body in ${\displaystyle \mathbf {R} ^{n}}$ has a linear image which contains the unit cross-polytope (the unit ball for the ${\displaystyle \ell _{1}^{n}}$ norm) and is contained in the unit cube (the unit ball for the ${\displaystyle \ell _{\infty }^{n}}$ norm).

By induction on the dimension ${\displaystyle n}$. Pick an arbitrary unit vector ${\displaystyle e_{n}\in V}$. Because the set of norm-1 points make up a convex symmetric body in ${\displaystyle V}$, there exists a hyperplane ${\displaystyle P_{n}}$ supporting ${\displaystyle V}$ at ${\displaystyle e_{n}}$. This is a consequence of the hyperplane separation theorem, which is a consequence of the Hahn–Banach theorem.

Now, define the dual vector ${\displaystyle e^{n}\in V^{*}}$, such that ${\displaystyle \{x\in V:e^{n}(x)=1\}=P_{n}}$. That is, the contour surfaces of ${\displaystyle e^{n}}$ are parallel to ${\displaystyle P_{n}}$.

Then, the subspace ${\displaystyle \ker(e^{n})}$ is a normed space of dimension ${\displaystyle n-1}$, and apply induction.

The lemma has a corollary with implications to approximation theory.

Let ${\displaystyle V}$ be an ${\displaystyle n}$-dimensional subspace of a normed vector space ${\displaystyle (X,\|\cdot \|)}$. Then there exists a projection ${\displaystyle P}$ of ${\displaystyle X}$ onto ${\displaystyle V}$ such that ${\displaystyle \|P\|\leq n}$.

Let ${\displaystyle \{e_{1},\dots ,e_{n}\}}$ be an Auerbach basis of ${\displaystyle V}$ and ${\displaystyle \{e^{1},\dots ,e^{n}\}}$ corresponding dual basis. By the Hahn–Banach theorem each ${\displaystyle e^{i}}$ extends to ${\displaystyle f^{i}\in X^{*}}$ such that ${\displaystyle \|f^{i}\|=1}$. Now set ${\displaystyle P(x)=\sum f^{i}(x)e_{i}}$. It is easy to check that ${\displaystyle P}$ is indeed a projection onto ${\displaystyle V}$ and that ${\displaystyle \|P\|\leq n}$ (this follows from the triangle inequality).