Normed spaces

Inherent

  • Topological space
  • Metric space
  • Vector space

Requirement

  • Valued Field
  • Vector space
  • Norm

Definition

Normed space

{\huge (\textcolor{Salmon}{X},\|\cdot \|) }
A normed space is a vector space over any valued field where a norm is defined.

Example

Real number line

The real line together with the euklidiean norm is a normed vector space.

(\R,\|\cdot\|_2) \hspace{1cm} \| \textcolor{Maroon}{x}\| := |\textcolor{Maroon}{x}| \hspace{1cm} \| \textcolor{Maroon}{x}  - \textcolor{Maroon}{y} \| := | \textcolor{Maroon}{x} - \textcolor{Maroon}{y}|

Induced Definition

Metric

{\huge (\textcolor{Salmon}{X},\|\cdot \| , \textcolor{RoyalBlue}{d} )  \hspace{1cm} \textcolor{RoyalBlue}{d}(\textcolor{Maroon}{x},\textcolor{Maroon}{y}) := \| \textcolor{Maroon}{x} - \textcolor{Maroon}{y} \| }
A normed space is also a metric space by the induced metric.

Induced Definition

Topology

{\huge (\textcolor{Salmon}{X},\|\cdot \| , \textcolor{violet}{\mathcal{T}}) \hspace{1cm} \textcolor{violet}{\mathcal{T}} := \{  B_r(\textcolor{Maroon}{x})\} \hspace{1cm} B_r(\textcolor{Maroon}{x}) := \{ \textcolor{Maroon}{y} \in \textcolor{Salmon}{X} : \textcolor{RoyalBlue}{d}(\textcolor{Maroon}{x},\textcolor{Maroon}{y}) < r \}} 
A normed space is also a topological space by the topology induced by the open balls.
The \ open \ balls \ are \ open.