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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 ...

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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-dimensionalnormed vector spaces.

Statement

Let (V,‖⋅‖){\displaystyle (V,\|\cdot \|)} be an n{\displaystyle n}-dimensional normed vector space. Then there exists a basis {e1,…,en}{\displaystyle \{e_{1},\dots,e_{n}\}} of V{\displaystyle V} such that ‖ei‖=1{\displaystyle \|e_{i}\|=1} and ‖ei‖=1{\displaystyle \|e^{i}\|=1} for i=1,…,n{\displaystyle i=1,\dots ,n}, where {e1,…,en}{\displaystyle \{e^{1},\dots,e^{n}\}} is a basis of V∗{\displaystyle V^{*}} dual to {e1,…,en}{\displaystyle \{e_{1},\dots,e_{n}\}}, i.e. ei(ej)=δij{\displaystyle e^{i}(e_{j})=\delta _{ij}}.

A basis with this property is called an Auerbach basis.

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

Geometric formulation

An equivalent statement is the following: any centrally symmetric convex body in Rn{\displaystyle \mathbf {R} ^{n}} has a linear image which contains the unit cross-polytope (the unit ball for the ℓ1n{\displaystyle \ell _{1}^{n}} norm) and is contained in the unit cube (the unit ball for the ℓ∞n{\displaystyle \ell _{\infty }^{n}} norm).

Proof

By induction on the dimension n{\displaystyle n}. Pick an arbitrary unit vector en∈V{\displaystyle e_{n}\in V}. Because the set of norm-1 points make up a convex symmetric body in V{\displaystyle V}, there exists a hyperplane Pn{\displaystyle P_{n}} supporting V{\displaystyle V} at en{\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 en∈V∗{\displaystyle e^{n}\en V^{*}}, such that {x∈V:en(x)=1}=Pn{\displaystyle \{x\in V:e^{n}(x)=1\}=P_{n}}. That is, the contour surfaces of en{\displaystyle e^{n}} are parallel to Pn{\displaystyle P_{n}}.

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

Corollary

The lemma has a corollary with implications to approximation theory.

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

Proof

Let {e1,…,en}{\displaystyle \{e_{1},\dots,e_{n}\}} be an Auerbach basis of V{\displaystyle V} and {e1,…,en}{\displaystyle \{e^{1},\dots,e^{n}\}} corresponding dual basis. By the Hahn–Banach theorem each ei{\displaystyle e^{i}} extends to fi∈X∗{\displaystyle f^{i}\in X^{*}} such that ‖fi‖=1{\displaystyle \|f^{i}\|=1}. Now set P(x)=∑fi(x)ei{\displaystyle P(x)=\sum f^{i}(x)e_{i}}. It is easy to check that P{\displaystyle P} is indeed a projection onto V{\displaystyle V} and that ‖P‖≤n{\displaystyle \|P\|\leq n} (this follows from the triangle inequality).

See also

References

  • Diestel, Joe; Jarchow, Hans; Tonge, Andrew (1995). Absolutely summing operators. Cambridge studies in advanced mathematics. Cambridge; New York: Cambridge University Press. p. 146. ISBN 978-0-521-43168-2.
  • Lindenstrauss, Joram; Tzafriri, Lior (1996). Classical Banach Spaces I and II: Sequence Spaces; Function Spaces. Springer. p. 16. ISBN 3540606289.
  • Meise, Reinhold; Vogt, Dietmar (1992). Einführung in die Funktionalanalysis[Introduction to Functional Analysis] (in German). Braunschweig: Vieweg. ISBN 3-528-07262-8.
  • Wojtaszczyk, Przemysław (1991). Banach spaces for analysts. Cambridge Studies in Advanced Mathematics. Vol. 25. Cambridge University Press. p. 75. ISBN 978-0521566759.