equivalent norms

Let x and x be two norms on a vector spaceMathworldPlanetmath V. These norms are equivalent norms if there exists a number C>1 such that

1CxxCx (1)

for all xV.

Since equation (1) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to

1CxxCx (2)

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 c,d such that


However, this condition is equivalent to the above by setting C=max{1/c,d}.

Some key results are as follows:

  1. 1.

    If γ>0 and x=γx, then and are equivalent. For example, if γ>1, then condition (1) holds with C=γ, and for γ<1, condition (2) holds with C=1/γ.

  2. 2.

    Suppose norms and are equivalent norms as in equation (1), and let Br(x) and Br(x) be the open balls with respect to and , respectively. By this result (http://planetmath.org/ScalingOfTheOpenBallInANormedVectorSpace) it follows that


    It follows that the identity mapMathworldPlanetmath from (V,) to (V,) is a homeomorphism. Or, alternatively, equivalent norms on V induce the same topology on V.

  3. 3.

    The converseMathworldPlanetmath of the last paragraph is also true, i.e. if two norms induce the same topology on V then they are equivalent. This follows from the fact that every continuousMathworldPlanetmathPlanetmath linear functionMathworldPlanetmath between two normed vector spacesPlanetmathPlanetmath is bounded (http://planetmath.org/BoundedOperator) (see this entry (http://planetmath.org/BoundedOperator)).

  4. 4.

    Suppose , and , are inner productMathworldPlanetmath. Suppose further that the induced norms and are equivalent as in equation 1. Then, by the polarization identityPlanetmathPlanetmath, the inner products satisfy

  5. 5.

    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 sphereMathworldPlanetmath 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)).

    It follows that on a finite dimensional vector space, one can check continuity and convergence with respect with any norm. If a sequence converges in one norm, it converges in all norms. In matrix analysis this is particularly useful as one can choose the norm that is most easily calculated.

  6. 6.

    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 C>0 such that

Title equivalent norms
Canonical name EquivalentNorms
Date of creation 2013-03-22 13:39:28
Last modified on 2013-03-22 13:39:28
Owner matte (1858)
Last modified by matte (1858)
Numerical id 10
Author matte (1858)
Entry type Definition
Classification msc 46B99