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 ${\displaystyle (V,\|\cdot \|)}$ ${\displaystyle x,y\in V}$

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

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 ${\displaystyle V}$

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

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.

literature

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