Diese Seite beinhaltet die Vektorräume, die in dem Kurs verwendet werden.
Sei
K
{\displaystyle \mathbb {K} }
V
=
(
V
,
+
)
{\displaystyle V=(V,+)}
V
{\displaystyle V}
K
{\displaystyle \mathbb {K} }
⋅
:
K
×
V
→
V
{\displaystyle \cdot :\mathbb {K} \times V\rightarrow V}
(
λ
,
v
)
↦
λ
⋅
v
{\displaystyle (\lambda ,v)\mapsto \lambda \cdot v}
definiert ist, die die folgenden Axiome erfüllt
λ
,
μ
∈
K
{\displaystyle \lambda ,\mu \in \mathbb {K} }
v
,
w
∈
V
{\displaystyle v,w\in V}
(ES)
1
⋅
v
=
v
{\displaystyle 1\cdot v=v}
(AMS)
λ
⋅
(
μ
⋅
v
)
=
(
λ
⋅
μ
)
⋅
v
.
{\displaystyle \lambda \cdot (\mu \cdot v)=(\lambda \cdot \mu )\cdot v.}
(DV)
λ
⋅
(
v
+
w
)
=
λ
⋅
v
+
λ
⋅
w
.
{\displaystyle \lambda \cdot (v+w)=\lambda \cdot v+\lambda \cdot w.}
(DS)
(
λ
+
μ
)
⋅
v
=
λ
⋅
v
+
μ
⋅
v
.
{\displaystyle (\lambda +\mu )\cdot v=\lambda \cdot v+\mu \cdot v.}
(Unterscheidung von Verknüpfungen -
K
{\displaystyle \mathbb {K} }
Durch welche Eigenschaften unterscheiden sich ein
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
Multiplikation im Körper
K
{\displaystyle \mathbb {K} }
Multiplikation mit Skalaren als äußere Verknüpfung,
Multiplikation von Elementen aus dem Vektorraum als innere Verknüpfung in einer
K
{\displaystyle \mathbb {K} }
(Unterscheidung von Verknüpfungen - Hilbert-Raum) Sei
K
:=
R
{\displaystyle \mathbb {K} :=\mathbb {R} }
K
:=
C
{\displaystyle \mathbb {K} :=\mathbb {C} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
K
{\displaystyle \mathbb {K} }
Bearbeiten
Seien
m
,
n
∈
N
{\displaystyle m,n\in \mathbb {N} }
M
a
t
(
m
×
n
,
Q
)
{\displaystyle Mat(m\times n,\mathbb {Q} )}
m
×
n
{\displaystyle m\times n}
Q
{\displaystyle \mathbb {Q} }
Q
{\displaystyle \mathbb {Q} }
m
⋅
n
{\displaystyle m\cdot n}
M
a
t
(
m
×
n
,
R
)
{\displaystyle Mat(m\times n,\mathbb {R} )}
m
×
n
{\displaystyle m\times n}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
m
⋅
n
{\displaystyle m\cdot n}
M
a
t
(
m
×
n
,
C
)
{\displaystyle Mat(m\times n,\mathbb {C} )}
m
×
n
{\displaystyle m\times n}
C
{\displaystyle \mathbb {C} }
C
{\displaystyle \mathbb {C} }
m
⋅
n
{\displaystyle m\cdot n}
Bearbeiten
Sei
C
(
[
a
,
b
]
,
K
)
{\displaystyle {\mathcal {C}}([a,b],\mathbb {K} )}
[
a
.
b
]
{\displaystyle [a.b]}
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
C
(
[
a
,
b
]
,
Q
)
{\displaystyle {\mathcal {C}}([a,b],\mathbb {Q} )}
Q
{\displaystyle \mathbb {Q} }
C
(
[
a
,
b
]
,
R
)
{\displaystyle {\mathcal {C}}([a,b],\mathbb {R} )}
R
{\displaystyle \mathbb {R} }
C
(
[
a
,
b
]
,
C
)
{\displaystyle {\mathcal {C}}([a,b],\mathbb {C} )}
C
{\displaystyle \mathbb {C} }
Bearbeiten
Mit
V
=
C
(
[
a
,
b
]
,
K
)
{\displaystyle V={\mathcal {C}}([a,b],\mathbb {K} )}
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
+
:
V
×
V
→
V
{\displaystyle +:V\times V\to V}
(
f
,
g
)
↦
f
+
g
:=
h
{\displaystyle (f,g)\mapsto f+g:=h}
h
(
x
)
:=
f
(
x
)
+
g
(
x
)
{\displaystyle h(x):=f(x)+g(x)}
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
Die äußere Verknüfung ist ebenfalls durch die Multiplikation der Funktionswerte mit dem Skalar argumentweise für jedes
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
⋅
:
K
×
V
→
V
{\displaystyle \cdot :\mathbb {K} \times V\to V}
(
λ
,
f
)
↦
λ
⋅
f
:=
h
{\displaystyle (\lambda ,f)\mapsto \lambda \cdot f:=h}
h
(
x
)
:=
λ
⋅
f
(
x
)
{\displaystyle h(x):=\lambda \cdot f(x)}
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
Die Kompaktheit des Definitionsbereiches
[
a
,
b
]
{\displaystyle [a,b]}
C
(
[
a
,
b
]
,
R
)
{\displaystyle {\mathcal {C}}([a,b],\mathbb {R} )}
[
a
,
b
]
{\displaystyle [a,b]}
R
{\displaystyle \mathbb {R} }
‖
f
‖
:=
∫
a
b
|
f
(
x
)
|
d
x
{\displaystyle \|f\|:=\displaystyle \int _{a}^{b}|f(x)|\,dx}
zu einem normierten Vektorraum (siehe auch Normen, Metriken, Topologie ). Mit den Halbnormen
‖
f
‖
n
:=
∫
−
n
+
n
|
f
(
x
)
|
d
x
{\displaystyle \|f\|_{n}:=\displaystyle \int _{-n}^{+n}|f(x)|\,dx}
wird
(
C
(
R
,
R
)
,
‖
⋅
‖
N
{\displaystyle ({\mathcal {C}}(\mathbb {R} ,\mathbb {R} ),\|\cdot \|_{\mathbb {N} }}
Bearbeiten
Sei
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
K
N
:=
{
(
a
n
)
n
∈
N
|
a
n
∈
K
für alle
n
∈
N
}
{\displaystyle \mathbb {K} ^{\mathbb {N} }:=\{(a_{n})_{n\in \mathbb {N} }\,|\,a_{n}\in \mathbb {K} \,{\mbox{ für alle }}n\in \mathbb {N} \}}
K
{\displaystyle \mathbb {K} }
c
o
o
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∃
n
0
∈
N
∀
n
≥
n
0
:
a
n
=
0
}
{\displaystyle c_{oo}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{n_{0}\in \mathbb {N} }\forall _{n\geq n_{0}}\,:\,a_{n}=0\}}
K
{\displaystyle \mathbb {K} }
c
o
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
lim
n
→
∞
a
n
=
0
}
{\displaystyle c_{o}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\lim _{n\to \infty }a_{n}=0\}}
c
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∃
a
0
∈
K
:
lim
n
→
∞
a
n
=
a
0
}
{\displaystyle c(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{a_{0}\in \mathbb {K} }\,:\,\lim _{n\to \infty }a_{n}=a_{0}\}}
K
{\displaystyle \mathbb {K} }
Bearbeiten
Sei
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
ℓ
1
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∑
n
=
1
∞
|
a
n
|
<
∞
}
{\displaystyle \ell _{1}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\sum _{n=1}^{\infty }|a_{n}|<\infty \}}
K
{\displaystyle \mathbb {K} }
ℓ
1
(
K
)
{\displaystyle \ell _{1}(\mathbb {K} )}
‖
a
‖
:=
∑
n
=
1
∞
|
a
n
|
{\displaystyle \|a\|:=\sum _{n=1}^{\infty }|a_{n}|}
ℓ
p
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∑
n
=
1
∞
|
a
n
|
p
<
∞
}
{\displaystyle \ell _{p}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\sum _{n=1}^{\infty }|a_{n}|^{p}<\infty \}}
K
{\displaystyle \mathbb {K} }
1
≤
p
<
∞
{\displaystyle 1\leq p<\infty }
0
<
p
<
1
{\displaystyle 0<p<1}
d
p
(
(
a
n
)
n
,
(
b
n
)
n
)
:=
∑
n
=
1
∞
|
a
n
−
b
n
|
p
{\displaystyle d_{p}((a_{n})_{n},(b_{n})_{n}):=\sum _{n=1}^{\infty }|a_{n}-b_{n}|^{p}}
ℓ
∞
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∃
C
>
0
:
sup
n
∈
N
|
a
n
|
<
C
}
{\displaystyle \ell _{\infty }(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{C>0}\,:\,\sup _{n\in \mathbb {N} }|a_{n}|<C\}}
K
{\displaystyle \mathbb {K} }
Bearbeiten
Sei
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
(
p
n
)
n
∈
N
{\displaystyle (p_{n})_{n\in \mathbb {N} }}
monoton fallende Nullfolge mit
0
<
p
n
≤
1
{\displaystyle 0<p_{n}\leq 1}
n
∈
N
{\displaystyle n\in \mathbb {N} }
ℓ
(
K
,
(
p
n
)
n
∈
N
)
:=
{
(
a
k
)
k
∈
N
∈
K
N
|
∑
k
=
1
∞
|
a
k
|
p
k
<
∞
}
{\displaystyle \ell (\mathbb {K} ,(p_{n})_{n\in \mathbb {N} }):=\{(a_{k})_{k\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\sum _{k=1}^{\infty }|a_{k}|^{p_{k}}<\infty \}}
K
{\displaystyle \mathbb {K} }
(
|
a
k
|
p
k
)
k
∈
N
{\displaystyle \left(|a_{k}|^{p_{k}}\right)_{k\in \mathbb {N} }}
Auf
ℓ
(
K
,
(
p
n
)
n
∈
N
)
{\displaystyle \ell (\mathbb {K} ,(p_{n})_{n\in \mathbb {N} })}
p
{\displaystyle p}
‖
a
‖
n
=
∑
k
=
1
∞
|
a
k
|
p
n
{\displaystyle \|a\|_{n}=\sum _{k=1}^{\infty }|a_{k}|^{p_{n}}}
a
=
(
a
k
)
k
∈
N
.
{\displaystyle a=(a_{k})_{k\in \mathbb {N} .}}
ℓ
(
K
,
(
p
n
)
n
∈
N
)
{\displaystyle \ell (\mathbb {K} ,(p_{n})_{n\in \mathbb {N} })}
p
{\displaystyle p}
‖
⋅
‖
N
{\displaystyle \|\cdot \|_{\mathbb {N} }}
Man beachte, dass für die
p
{\displaystyle p}
n
{\displaystyle n}
k
∈
N
{\displaystyle k\in \mathbb {N} }
Sei
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
(
V
,
‖
⋅
‖
{\displaystyle (V,\|\cdot \|}
K
{\displaystyle \mathbb {K} }
V
[
x
]
:=
{
p
|
(
p
n
)
n
∈
N
∈
c
o
o
(
V
)
∧
p
(
x
)
:=
∑
n
=
0
∞
p
n
⋅
x
n
}
{\displaystyle V[x]:=\left\{p\,\left|\,(p_{n})_{n\in \mathbb {N} }\in c_{oo}(V)\wedge p(x):=\sum _{n=0}^{\infty }p_{n}\cdot x^{n}\right.\right\}}
V
{\displaystyle V}
Für ein spezielles
x
∈
K
{\displaystyle x\in \mathbb {K} }
p
(
x
)
∈
V
{\displaystyle p(x)\in V}
V
{\displaystyle V}
x
n
∈
K
{\displaystyle x^{n}\in \mathbb {K} }
x
∈
K
{\displaystyle x\in \mathbb {K} }
Bearbeiten
Die innere und äußere Verknüpfungen auf den Folgenräumen sind jeweils komponentenweise definiert, analog zur Vektorraum Addition und Multiplikation mit Skalaren im
Q
n
{\displaystyle \mathbb {Q} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
n
{\displaystyle \mathbb {C} ^{n}}
V
=
K
N
{\displaystyle V=\mathbb {K} ^{\mathbb {N} }}
K
=
Q
,
R
,
C
)
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} )}
a
:=
(
a
n
)
n
∈
N
{\displaystyle a:=(a_{n})_{n\in \mathbb {N} }}
b
:=
(
b
n
)
n
∈
N
{\displaystyle b:=(b_{n})_{n\in \mathbb {N} }}
c
:=
(
c
n
)
n
∈
N
{\displaystyle c:=(c_{n})_{n\in \mathbb {N} }}
+
:
V
×
V
→
V
{\displaystyle +:V\times V\to V}
(
a
,
b
)
↦
a
+
b
:=
c
{\displaystyle (a,b)\mapsto a+b:=c}
c
n
:=
a
n
+
b
n
{\displaystyle c_{n}:=a_{n}+b_{n}}
n
∈
N
{\displaystyle n\in \mathbb {N} }
Die äußere Verknüpfung ist ebenfalls durch die Multiplikation der Komponenten der Folge mit dem Skalar definert:
⋅
:
K
×
V
→
V
{\displaystyle \cdot :\mathbb {K} \times V\to V}
(
λ
,
a
)
↦
λ
⋅
a
:=
c
{\displaystyle (\lambda ,a)\mapsto \lambda \cdot a:=c}
c
n
:=
λ
⋅
a
n
{\displaystyle c_{n}:=\lambda \cdot a_{n}}
n
∈
N
{\displaystyle n\in \mathbb {N} }
Betrachten Sie den Zahlenmenge der reellen Zahlen
R
{\displaystyle \mathbb {R} }
Q
{\displaystyle \mathbb {Q} }
(
R
,
+
,
⋅
,
Q
)
{\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}
Q
{\displaystyle \mathbb {Q} }
Zeigen Sie, dass die von
v
1
=
3
{\displaystyle v_{1}=3}
v
2
=
3
{\displaystyle v_{2}={\sqrt {3}}}
U
{\displaystyle U}
Q
{\displaystyle \mathbb {Q} }
(
R
,
+
,
⋅
,
Q
)
{\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}
5
⋅
Q
{\displaystyle {\sqrt {5}}\cdot \mathbb {Q} }
0
{\displaystyle 0}
Analysieren Sie die Mengeninklusion bei den Folgenräumen und betrachtet Sie auch die Konvergenzeigenschaften von Reihen, die aus den Folgen generiert werden können mit:
ℓ
p
(
K
)
:=
{
(
x
n
)
n
∈
K
N
:
∑
n
=
1
∞
|
x
n
|
p
<
∞
}
{\displaystyle \quad \ell ^{p}(\mathbb {K} ):=\left\{(x_{n})_{n}\in \mathbb {K} ^{\mathbb {N} }\,:\,\sum _{n=1}^{\infty }|x_{n}|^{p}<\infty \right\}}
Welche Teilmengenbeziehung besteht zwischen
ℓ
1
(
K
)
{\displaystyle \ell ^{1}(\mathbb {K} )}
c
o
(
K
)
{\displaystyle c_{o}(\mathbb {K} )}
ℓ
1
(
V
)
{\displaystyle \ell ^{1}(V)}
c
o
(
V
)
{\displaystyle c_{o}(V)}
(
V
,
‖
⋅
‖
)
{\displaystyle (V,\|\cdot \|)}
(
V
,
d
)
{\displaystyle (V,d)}
![{\displaystyle {\mathcal {C}}([a,b],\mathbb {K} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a56a08fb8218f574e8a6bb91b6b07013271cecf)
![{\displaystyle [a.b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1c1a8e2358869e1a65d39ab7f1717d0f2c55f3)
![{\displaystyle {\mathcal {C}}([a,b],\mathbb {Q} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3fbc319232d1f361f01144fe7bb8c8af3b28b45)
![{\displaystyle {\mathcal {C}}([a,b],\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f380817b9ebaf6a6afb1036f63f41b10311a9b)
![{\displaystyle {\mathcal {C}}([a,b],\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10966b1db37d405ee18bda34ca860aa1d8ee9adc)
![{\displaystyle V={\mathcal {C}}([a,b],\mathbb {K} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b51b930f2ddd56748dde5e6b2592747af837929)
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle V[x]:=\left\{p\,\left|\,(p_{n})_{n\in \mathbb {N} }\in c_{oo}(V)\wedge p(x):=\sum _{n=0}^{\infty }p_{n}\cdot x^{n}\right.\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/339da3568a27c9270bb56c248bfaf23dc2e0b267)
