# 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

Relationships between norm, metric and topology

If a normalized vector space , the norm of the space induces a metric by forming the difference between two vectors${\ displaystyle (V, \ | \ cdot \ |)}$${\ displaystyle x, y \ in V}$

${\ displaystyle d (x, y): = \ | xy \ |}$.

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 ${\ displaystyle V}$ ${\ displaystyle (V, d)}$${\ displaystyle x \ in V}$

${\ displaystyle U _ {\ varepsilon} (x): = \ {\, y \ in V, \, d \, (x, y) <\ varepsilon \, \}}$

define. A subset is then called open , if ${\ displaystyle M \ subset V}$

${\ displaystyle \ forall \ {x \ in M} \; {\ exists \ \ varepsilon}> 0: U _ {\ varepsilon} (x) \ subset M}$

applies. The metric now induces a topology via these open sets${\ displaystyle V}$

${\ displaystyle {\ mathcal {T}}: = \ {M \ subset V, \, M \, {\ text {open}} \}}$.

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. ${\ displaystyle (V, {\ mathcal {T}})}$

## 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.${\ displaystyle x}$${\ displaystyle V}$

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.${\ displaystyle M_ {1}, \ ldots, M_ {n}}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle \ varepsilon _ {1}, \ ldots, \ varepsilon _ {n}}$${\ displaystyle x}$${\ displaystyle U _ {\ varepsilon _ {i}} (x) \ subset M_ {i}}$${\ displaystyle i = 1, \ ldots, n}$${\ displaystyle \ varepsilon = \ min \ {\ varepsilon _ {1}, \ ldots, \ varepsilon _ {n} \}}$${\ displaystyle U _ {\ varepsilon} (x) \ subset M_ {1} \ cap \ ldots \ cap M_ {n}}$

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.${\ displaystyle I}$${\ displaystyle M_ {i}}$${\ displaystyle i \ in I}$${\ displaystyle x}$${\ displaystyle i \ in I}$${\ displaystyle x \ in M_ {i}}$${\ displaystyle \ varepsilon}$${\ displaystyle U _ {\ varepsilon} (x) \ subset M_ {i}}$${\ displaystyle U _ {\ varepsilon} (x) \ subset \ cup _ {i \ in I} M_ {i}}$

## 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.${\ displaystyle x, y \ in V}$${\ displaystyle x \ neq y}$${\ displaystyle U _ {\ varepsilon} (x)}$${\ displaystyle U _ {\ varepsilon} (y)}$${\ displaystyle \ textstyle \ varepsilon = {\ tfrac {1} {2}} d (x, y)}$
• 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.