Lineare Algebra 2/Gemischte Definitionsabfrage/22/Aufgabe/Lösung

${\displaystyle {}d\colon M\times M\rightarrow \mathbb {R} }$

${\displaystyle {}x,y,z\in M}$

1. ${\displaystyle {}d\left(x,y\right)=0\Longleftrightarrow x=y}$,
2. ${\displaystyle {}d\left(x,y\right)=d(y,x)}$, und
3. ${\displaystyle {}d\left(x,y\right)\leq d(x,z)+d(z,y)}$.

${\displaystyle \varphi \colon V\longrightarrow W}$

heißt antilinear, wenn

${\displaystyle {}\varphi (u+v)=\varphi (u)+\varphi (v)\,}$

für alle ${\displaystyle {}u,v\in V}$ und wenn

${\displaystyle {}\varphi (\lambda v)={\overline {\lambda }}\varphi (v)\,}$

für alle ${\displaystyle {}v\in V}$ und ${\displaystyle {}\lambda \in \mathbb {C} }$ gilt.

${\displaystyle {}F}$

${\displaystyle {}X_{1},\ldots ,X_{n}}$

${\displaystyle {}F=\sum _{\nu }c_{\nu }X^{\nu }\,}$

mit ${\displaystyle {}c_{\nu }\in K}$.

${\displaystyle {}\sim }$

${\displaystyle {}M}$

${\displaystyle {}x,y,z\in M}$

1. ${\displaystyle {}x\sim x}$.
2. Aus ${\displaystyle {}x\sim y}$ folgt ${\displaystyle {}y\sim x}$.
3. Aus ${\displaystyle {}x\sim y}$ und ${\displaystyle {}y\sim z}$ folgt ${\displaystyle {}x\sim z}$.

${\displaystyle {}A_{n}:={\left\{\sigma \in S_{n}\mid \operatorname {sgn} (\sigma )=1\right\}}\,.}$