Equivalent norms

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Equivalence of the Euclidean norm (blue) and the maximum norm (red) in two dimensions

In mathematics, equivalent norms are a pair of abstract distance concepts, so-called norms , which generate identical convergence concepts. A more detailed distinction is made between stronger norms (synonymously also called finer norms ) and weaker norms (synonymously also called coarser norms ) and two norms are called equivalent if they are both stronger and weaker than their counterpart.


Given is a vector space over (in most cases or ) on which two norms and are defined.

Then means stronger or finer than if there is a positive number such that

is. Correspondingly, weaker or coarser than is also mentioned.

The norms and are called equivalent if there are positive numbers such that

applies. Two norms are therefore equivalent if is stronger than and is stronger than .


Finite dimensional

Let it be given , provided with the maximum norm and the sum norm


Then it is always because of


So is


accordingly the maximum norm is stronger than the sum norm. It is always the other way around


since the entry with the largest amount in a vector is never greater than the sum of the amounts of all entries in the vector. Thus the sum norm is stronger than the maximum norm. Overall then applies


The maximum norm and the sum norm im are therefore equivalent. In fact it can be shown that all norms are equivalent on arbitrary finite-dimensional vector spaces.

Infinitely dimensional

If one considers the vector space of the real-valued continuous functions on the closed interval from zero to one, two norms can be defined:

  • On the one hand the supremum norm , which is well-defined due to the boundedness of continuous functions on the compact interval .
  • On the other hand, continuous functions are always measurable in this context and are contained in Lp space because of their limited nature . This means that the L1 standard

The integral can always be estimated upwards by the largest possible function value, so it applies here

and thus


The supreme norm is therefore stronger than the L1 norm.

However, the two standards are not equivalent: For example, the following applies to the functions defined by with and . So there can be no constant with for all functions in .


If two norms and are given and is stronger than , then is the sphere

always included in the sphere . A convergence with respect to always automatically forces a convergence with respect to , since the standard spheres of always contain the standard spheres of after rescaling . Thus always "majorizes" .

The equivalence of the norms now means that both is stronger than and that is stronger than . According to the above argument, a sequence converges with respect to exactly when it converges with respect to .


  • If the norm is stronger than , then applies to the generated metrics
that then is also stronger than .
  • The same applies: Is stronger than , the topology generated by is finer or stronger than the topology generated by.
  • In finite-dimensional vector spaces, all norms are equivalent.