# Equivalent norms

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.

## definition

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 .

## Examples

### 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

- define.

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 .

## interpretation

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 .

## properties

- 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.

## literature

- Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin / Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
- Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg / Dordrecht / London / New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .