Auerbach base

An Auerbach basis is a linearly independent subset of a normalized vector space with special properties.

definition

Let be a normalized vector space . A set is called an Auerbach basis of X if the following conditions are met: ${\displaystyle X}$ ${\displaystyle A\subseteq X}$

• The linear envelope of the set lies close in . ${\displaystyle A}$ ${\displaystyle X}$
• The following applies to each , with the closure of the linear envelope of the set . ${\displaystyle a\in A}$ ${\displaystyle \|a\|=\inf\{\|a-b\|:b\in [A\setminus \{a\}]\}}$ ${\displaystyle [B]}$ ${\displaystyle B}$
• The set is linearly independent . (This condition follows from the previous one; it must even apply to all of the relationship .) ${\displaystyle A}$ ${\displaystyle x\in A}$ ${\displaystyle x\notin [A\setminus \{x\}]}$

An Auerbach basis is called a normalized Auerbach basis if all vectors of the set have the norm 1. ${\displaystyle A}$ ${\displaystyle A}$

Motivation and History

The equation holds in every finite-dimensional Hilbert space

${\displaystyle \|x\|=\inf\{\|x-y\|:y\in [B]\}}$

exactly when the vector is normal to the subspace generated by . In this sense, the concept of the normalized Auerbach basis is a generalization of the concept of the orthonormal basis . ${\displaystyle x}$ ${\displaystyle B}$

This term was defined in Herman Auerbach's dissertation . The dissertation itself, which was written in 1929, is considered lost. However, it is mentioned in a monograph by Stefan Banach from 1932.

Equivalent Definitions

In a Banach space X , a set A of vectors is a normalized Auerbach basis if and only if the following conditions hold:

1. ${\displaystyle [A]=X}$.
2. Applies to everyone ${\displaystyle a\in A}$ ${\displaystyle a\notin [A\setminus \{a\}]}$
3. The normalization condition applies to each ${\displaystyle a\in A}$ ${\displaystyle \|a\|=1}$
4. There are a set of continuous linear functionals on (i.e. a subset of the topological dual space ) with the properties ${\displaystyle \{f_{a}:a\in A\}}$ ${\displaystyle X}$
   • ${\displaystyle f_{a}(b)=\delta _{ab}}$for everyone . This is the Kronecker Delta . ${\displaystyle a,b\in A}$ ${\displaystyle \delta _{ab}}$
   • ${\displaystyle \|f_{a}\|=1}$for everyone . ${\displaystyle a\in A}$

Hahn-Banach's theorem is used to prove it .

For vector spaces of finite dimension, the conditions 1 + 2 simply mean that A is a basis . In finite-dimensional normed vector spaces, the Auerbach lemma says that there is always an Auerbach basis.

literature

• Herman Auerbach: O polu krzywych wypukłych o średnicach sprzężonych (On surfaces of convex curves with conjugate diameters), dissertation at the University of Lwów (1929; in Polish).
• Stefan Banach : Théorie des opérations linéaires. Monograph matematyczne , edited by M. Garasiński, Warsaw 1932.
• Bartoszyński et al .: On bases in Banach spaces . Studia Math. 170 (2005), no.2, 147--171.
• Dirk Werner : Functional Analysis . 5th edition. Springer, Berlin Heidelberg / New York 2005, ISBN 3540213813