Auerbach's lemma

The lemma of Auerbach (after Herman Auerbach ) is a statement of functional analysis. It states that in an n- dimensional normed vector space always exists an Auerbach basis. The quantity E is called an Auerbach basis of E if exist in the dual space of E with norm 1, such that for all. It is the Kronecker delta, that is equal to 1 if j = k, and equal to 0 otherwise

Because of the equations, the vectors are linearly independent, so they form a basis of the vector space. The proof uses tools from linear algebra and elementary calculus.

In the case of the Euclidean norm on a finite dimensional vector space or the unit vectors satisfy the statement of the lemma. The lemma of Auerbach also makes a statement about an arbitrary vector space norm, and is then not as obvious as the case of the Euclidean vector space.

In Hilbert spaces, each orthonormal basis is an Auerbach basis. As in the above lemma, take the functionals. In some situations, as in the following application, an Auerbach basis can act as a substitute for orthonormal bases.

Application

The following statement about not necessarily finite-dimensional spaces shows how this lemma can be used.

If E is a normed space and F an n-dimensional subspace, then there's a continuous projection P from E to F.

By the lemma of the n - dimensional subspace F has a base with Auerbach and by the theorem of Hahn- Banach there with and. By recalculating it can be shown then that

A projection from E to F is.

This rate can be significantly improved, there are according to the rate of Kadec - Snobar even projections with norm less than or equal, but the proof of this statement is much more difficult.