for all .
it follows that the definition is well defined. In other words, and are equivalent if and only if and are equivalent. An alternative condition is that there exist positive real numbers such that
However, this condition is equivalent to the above by setting .
Some key results are as follows:
Suppose norms and are equivalent norms as in equation (1), and let and be the open balls with respect to and , respectively. By this result (http://planetmath.org/ScalingOfTheOpenBallInANormedVectorSpace) it follows that
The converse of the last paragraph is also true, i.e. if two norms induce the same topology on then they are equivalent. This follows from the fact that every continuous linear function between two normed vector spaces is bounded (http://planetmath.org/BoundedOperator) (see this entry (http://planetmath.org/BoundedOperator)).
On a finite dimensional vector space all norms are equivalent (see this page (http://planetmath.org/ProofThatAllNormsOnFiniteVectorSpaceAreEquivalent)). This is easy to understand as the unit sphere is compact if and only if a space is finite dimensional. On infinite dimensional spaces this result does not hold (see this page (http://planetmath.org/AllNormsAreNotEquivalent)).
The concept of equivalent norms also generalize to possibly non-symmetric norms. In this setting, all norms are also equivalent on a finite dimensional vector space. In particular, and are equivalent, and there exists such that
|Date of creation||2013-03-22 13:39:28|
|Last modified on||2013-03-22 13:39:28|
|Last modified by||matte (1858)|