# Standard topology

In mathematics, a **norm topology** is a topology on a normalized vector space that was induced by the norm of the vector space .

## definition

If a normalized vector space , the norm of the space induces a metric by forming the difference between two vectors

- .

on . With this metric, the vector space becomes a metric space . A metric can now be used to a ε-environment to a vector by

define. A subset is then called open , if

applies. The metric now induces a topology via these open sets

- .

With this topology, the vector space becomes a topological vector space and this topology, which is ultimately induced by the norm, is called the norm topology.

## Topology axioms

The norm topology is actually a topology, as can be demonstrated by checking the three topology axioms , which are valid in the following form for all metric spaces.

1. The empty set and the basic set are open:

- The empty set is open since there is none for which a suitable ε-neighborhood has to be found. The basic set is open because it is an ε-neighborhood of all its elements.

2. The intersection of finitely many open sets is open:

- Be the sets with open. Then there are bounds and one from the intersection of these sets such that for holds. If you choose now , then the average of these sets is open.

3. The union of any number of open sets is open:

- Let now an arbitrary index set and are the amounts for open. If this is the union of these sets, then there is an index with and a bound so that holds. From this it follows and thus the union of these sets is open.

## properties

- The standard topology is a special strong topology . It is to be distinguished from the weak topology and the weak - * - topology .
- A flask equipped with a standard topology topological space is always Hausdorff because two vectors with by environments and with are separated.
- According to Kolmogoroff's criterion for normalization , the topology of a Hausdorff topological vector space is generated by a norm if and only if it has a bounded and convex null neighborhood.

## literature

- Dirk Werner : Functional Analysis . Springer-Verlag, 2007, ISBN 978-3-540-72533-6 .