Es sei
K
∈
{
R
,
C
}
{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}
V
{\displaystyle V}
K
{\displaystyle \mathbb {K} }
x
,
y
∈
V
{\displaystyle x,y\in V}
|
⟨
x
,
y
⟩
|
≤
⟨
x
,
x
⟩
⟨
y
,
y
⟩
{\displaystyle \left\vert \langle x,y\rangle \right\vert \leq {\sqrt {\langle x,x\rangle }}{\sqrt {\langle y,y\rangle }}}
Gleichheit liegt genau dann vor, wenn
x
,
y
{\displaystyle x,y}
Die Aussage ist für
x
=
0
{\displaystyle x=0}
x
≠
0
{\displaystyle x\neq 0}
⟨
x
,
x
⟩
>
0
{\displaystyle \langle x,x\rangle >0}
λ
∈
K
{\displaystyle \lambda \in \mathbb {K} }
⟨
λ
x
+
y
,
λ
x
+
y
⟩
=
⟨
λ
x
,
λ
x
⟩
+
⟨
λ
x
,
y
⟩
+
⟨
y
,
λ
x
⟩
+
⟨
y
,
y
⟩
=
|
λ
|
2
⟨
x
,
x
⟩
+
⟨
λ
x
,
y
⟩
+
⟨
λ
x
,
y
⟩
¯
+
⟨
y
,
y
⟩
=
|
λ
|
2
⟨
x
,
x
⟩
+
2
Re
(
λ
⟨
x
,
y
⟩
)
+
⟨
y
,
y
⟩
{\displaystyle {\begin{array}{lll}\langle \lambda x+y,\lambda x+y\rangle &=&\langle \lambda x,\lambda x\rangle +\langle \lambda x,y\rangle +\langle y,\lambda x\rangle +\langle y,y\rangle \\&=&|\lambda |^{2}\langle x,x\rangle +\langle \lambda x,y\rangle +{\overline {\langle \lambda x,y\rangle }}+\langle y,y\rangle \\&=&|\lambda |^{2}\langle x,x\rangle +2{\textrm {Re}}\left(\lambda \langle x,y\rangle \right)+\langle y,y\rangle \\\end{array}}}
sowie
|
λ
+
⟨
x
,
y
⟩
¯
⟨
x
,
x
⟩
|
2
=
(
λ
+
⟨
x
,
y
⟩
¯
⟨
x
,
x
⟩
)
⋅
(
λ
¯
+
⟨
x
,
y
⟩
⟨
x
,
x
⟩
)
=
λ
λ
¯
+
λ
⟨
x
,
y
⟩
⟨
x
,
x
⟩
+
λ
¯
⟨
x
,
y
⟩
¯
⟨
x
,
x
⟩
+
⟨
x
,
y
⟩
⟨
x
,
x
⟩
⟨
x
,
y
⟩
¯
⟨
x
,
x
⟩
=
|
λ
|
2
+
2
Re
(
λ
⟨
x
,
y
⟩
)
⟨
x
,
x
⟩
+
|
⟨
x
,
y
⟩
⟨
x
,
x
⟩
|
2
.
{\displaystyle {\begin{array}{lll}\left|\lambda +{\frac {\overline {\langle x,y\rangle }}{\langle x,x\rangle }}\right|^{2}&=&\left(\lambda +{\frac {\overline {\langle x,y\rangle }}{\langle x,x\rangle }}\right)\cdot \left({\overline {\lambda }}+{\frac {\langle x,y\rangle }{\langle x,x\rangle }}\right)\\&=&\lambda {\overline {\lambda }}+\lambda {\frac {\langle x,y\rangle }{\langle x,x\rangle }}+{\overline {\lambda }}{\frac {\overline {\langle x,y\rangle }}{\langle x,x\rangle }}+{\frac {\langle x,y\rangle }{\langle x,x\rangle }}{\frac {\overline {\langle x,y\rangle }}{\langle x,x\rangle }}\\&=&|\lambda |^{2}+2{\frac {{\textrm {Re}}\left(\lambda \langle x,y\rangle \right)}{\langle x,x\rangle }}+\left|{\frac {\langle x,y\rangle }{\langle x,x\rangle }}\right|^{2}\ .\\\end{array}}}
Dies impliziert für jedes
λ
∈
K
{\displaystyle \lambda \in \mathbb {K} }
⟨
λ
x
+
y
,
λ
x
+
y
⟩
⟨
x
,
x
⟩
=
|
λ
+
⟨
x
,
y
⟩
¯
⟨
x
,
x
⟩
|
2
+
⟨
y
,
y
⟩
⟨
x
,
x
⟩
−
|
⟨
x
,
y
⟩
⟨
x
,
x
⟩
|
2
{\displaystyle {\frac {\langle \lambda x+y,\lambda x+y\rangle }{\langle x,x\rangle }}=\left|\lambda +{\frac {\overline {\langle x,y\rangle }}{\langle x,x\rangle }}\right|^{2}+{\frac {\langle y,y\rangle }{\langle x,x\rangle }}-\left|{\frac {\langle x,y\rangle }{\langle x,x\rangle }}\right|^{2}}
welches eine reelle Zahl ist.
Daraus folgt
min
λ
∈
K
{
⟨
λ
x
+
y
,
λ
x
+
y
⟩
⟨
x
,
x
⟩
}
=
D
:=
⟨
y
,
y
⟩
⟨
x
,
x
⟩
−
|
⟨
x
,
y
⟩
⟨
x
,
x
⟩
|
2
{\displaystyle \min \limits _{\lambda \in \mathbb {K} }\left\{{\frac {\langle \lambda x+y,\lambda x+y\rangle }{\langle x,x\rangle }}\right\}=D:={\frac {\langle y,y\rangle }{\langle x,x\rangle }}-\left|{\frac {\langle x,y\rangle }{\langle x,x\rangle }}\right|^{2}}
Nun gilt
⟨
λ
x
+
y
,
λ
x
+
y
⟩
≥
0
{\displaystyle \langle \lambda x+y,\lambda x+y\rangle \geq 0}
λ
∈
K
{\displaystyle \lambda \in \mathbb {K} }
λ
∈
K
{\displaystyle \lambda \in \mathbb {K} }
x
,
y
{\displaystyle x,y}
D
>
0
{\displaystyle D>0}
D
=
0
{\displaystyle D=0}
D
>
0
⇔
|
⟨
x
,
y
⟩
|
<
⟨
x
,
x
⟩
⟨
y
,
y
⟩
{\displaystyle D>0\Leftrightarrow \left\vert \langle x,y\rangle \right\vert <{\sqrt {\langle x,x\rangle }}{\sqrt {\langle y,y\rangle }}}
und entsprechend
D
=
0
⇔
|
⟨
x
,
y
⟩
|
=
⟨
x
,
x
⟩
⟨
y
,
y
⟩
{\displaystyle D=0\Leftrightarrow \left\vert \langle x,y\rangle \right\vert ={\sqrt {\langle x,x\rangle }}{\sqrt {\langle y,y\rangle }}}
◻
{\displaystyle \Box }
