Auerbach's lemma

Template:Short description 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 $(V,\Vert \cdot \Vert )$ be an $n$-dimensional normed vector space. Then there exists a basis $\{e_1,\dots ,e_n\}$ of $V$ such that $\Vert e_i\Vert =1$ and $\Vert e^i\Vert =1$ for $i=1,\dots ,n$, where $\{e^1,\dots ,e^n\}$ is a basis of $V^{\ast }$ dual to $\{e_1,\dots ,e_n\}$, i.e. $e^i(e_j)={\delta }_{ij}$.

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 $\{(e_i|\cdot )\}$).

Geometric formulation

An equivalent statement is the following: any centrally symmetric convex body in ${𝐑}^n$ 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 ${\ell }_{\mathrm{\infty }}^n$ norm).

Proof

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

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

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

Corollary

The lemma has a corollary with implications to approximation theory.

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

Proof

Let $\{e_1,\dots ,e_n\}$ be an Auerbach basis of $V$ and $\{e^1,\dots ,e^n\}$ corresponding dual basis. By the Hahn–Banach theorem each $e^i$ extends to $f^i\in X^{\ast }$ such that $\Vert f^i\Vert =1$. Now set $P(x)=\sum f^i(x)e_i$. It is easy to check that $P$ is indeed a projection onto $V$ and that $\Vert P\Vert \le n$ (this follows from the triangle inequality).

See also

• Markushevich basis

References

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