Defining: vector space
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End-dimensional vector spaces 1
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''(Distinction between operations -
K
{\textstyle \mathbb {K} }
What are the characteristics of a
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
Multiplication in field
K
{\textstyle \mathbb {K} }
scalar multiplication as a binary function from
K
×
V
{\textstyle \mathbb {K} \times V}
V
{\textstyle V}
Multiplication of elements from vector space as an inner link in a
K
{\textstyle \mathbb {K} }
'(Multiplications - Hilbert space) Be
K
:=
R
{\textstyle \mathbb {K} :=\mathbb {R} }
K
:=
C
{\textstyle \mathbb {K} :=\mathbb {C} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
K
{\textstyle \mathbb {K} }
Finite dimensional vector spaces 2
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Be
m
,
n
∈
N
{\textstyle m,n\in \mathbb {N} }
M
a
t
(
m
×
n
,
Q
)
{\textstyle Mat(m\times n,\mathbb {Q} )}
m
×
n
{\textstyle m\times n}
Q
{\textstyle \mathbb {Q} }
Q
{\textstyle \mathbb {Q} }
m
⋅
n
{\textstyle m\cdot n}
M
a
t
(
m
×
n
,
R
)
{\textstyle Mat(m\times n,\mathbb {R} )}
m
×
n
{\textstyle m\times n}
R
{\textstyle \mathbb {R} }
R
{\textstyle \mathbb {R} }
m
⋅
n
{\textstyle m\cdot n}
M
a
t
(
m
×
n
,
C
)
{\textstyle Mat(m\times n,\mathbb {C} )}
m
×
n
{\textstyle m\times n}
C
{\textstyle \mathbb {C} }
C
{\textstyle \mathbb {C} }
m
⋅
n
{\textstyle m\cdot n}
Infinite-dimensional vector spaces of functions 1
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Be
C
(
[
a
,
b
]
,
K
)
{\textstyle {\mathcal {C}}([a,b],\mathbb {K} )}
[
a
.
b
]
{\textstyle [a.b]}
K
=
Q
,
R
,
C
{\textstyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
C
(
[
a
,
b
]
,
Q
)
{\textstyle {\mathcal {C}}([a,b],\mathbb {Q} )}
Q
{\textstyle \mathbb {Q} }
C
(
[
a
,
b
]
,
R
)
{\textstyle {\mathcal {C}}([a,b],\mathbb {R} )}
R
{\textstyle \mathbb {R} }
C
(
[
a
,
b
]
,
C
)
{\textstyle {\mathcal {C}}([a,b],\mathbb {C} )}
C
{\textstyle \mathbb {C} }
Internal and external link to vector spaces of functions 2
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The internal link is defined as follows:
+
:
V
×
V
→
V
{\displaystyle +:V\times V\to V}
(
f
,
g
)
↦
f
+
g
:=
h
{\textstyle (f,g)\mapsto f+g:=h}
h
(
x
)
:=
f
(
x
)
+
g
(
x
)
{\textstyle h(x):=f(x)+g(x)}
x
∈
[
a
,
b
]
{\textstyle x\in [a,b]}
The external feature is likewise defined by the multiplication of the function values with the scalar for each
x
∈
[
a
,
b
]
{\textstyle x\in [a,b]}
⋅
:
K
×
V
→
V
{\displaystyle \cdot :\mathbb {K} \times V\to V}
(
λ
,
f
)
↦
λ
⋅
f
:=
h
{\textstyle (\lambda ,f)\mapsto \lambda \cdot f:=h}
h
(
x
)
:=
λ
⋅
f
(
x
)
{\textstyle h(x):=\lambda \cdot f(x)}
x
∈
[
a
,
b
]
{\textstyle x\in [a,b]}
Vector Space of Continuous Functions
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The compactness of the definition range
[
a
,
b
]
{\textstyle [a,b]}
C
(
[
a
,
b
]
,
R
)
{\textstyle {\mathcal {C}}([a,b],\mathbb {R} )}
[
a
,
b
]
{\textstyle [a,b]}
R
{\textstyle \mathbb {R} }
‖
f
‖
:=
∫
a
b
|
f
(
x
)
|
d
x
{\displaystyle \|f\|:=\displaystyle \int _{a}^{b}|f(x)|\,dx}
to a standardized vector space (see also norms, metrics, topology ). With the semi-standards
‖
f
‖
n
:=
∫
−
n
+
n
|
f
(
x
)
|
d
x
{\displaystyle \|f\|_{n}:=\displaystyle \int _{-n}^{+n}|f(x)|\,dx}
becomes
(
C
(
R
,
R
)
,
‖
⋅
‖
N
{\textstyle ({\mathcal {C}}(\mathbb {R} ,\mathbb {R} ),\|\cdot \|_{\mathbb {N} }}
Infinite-dimensional vector spaces of sequences 3
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Be
K
=
Q
,
R
,
C
)
{\textstyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} )}
K
N
:=
{
(
a
n
)
n
∈
N
|
a
n
∈
K
für alle
n
∈
N
}
{\textstyle \mathbb {K} ^{\mathbb {N} }:=\{(a_{n})_{n\in \mathbb {N} }\,|\,a_{n}\in \mathbb {K} \,{\mbox{ für alle }}n\in \mathbb {N} \}}
K
{\textstyle \mathbb {K} }
c
o
o
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
∃
n
0
∈
N
∀
n
≥
n
0
:
a
n
=
0
}
{\textstyle 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
{\textstyle \mathbb {K} }
c
o
(
K
)
:=
{
(
a
n
)
n
∈
N
∈
K
N
|
lim
n
→
∞
a
n
=
0
}
{\textstyle 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
o
∈
K
:
lim
n
→
∞
a
n
=
a
o
}
{\textstyle c(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{a_{o}\in \mathbb {K} }\,:\,\lim _{n\to \infty }a_{n}=a_{o}\}}
K
{\textstyle \mathbb {K} }
Infinite-dimensional vector spaces of sequences 4
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Let
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}}
p
{\displaystyle p}
‖
(
a
n
)
n
‖
p
:=
∑
n
=
1
∞
|
a
n
|
p
{\displaystyle \|(a_{n})_{n}\|_{p}:=\sum _{n=1}^{\infty }|a_{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} }
ℓ
∞
(
K
)
{\displaystyle \ell _{\infty }(\mathbb {K} )}
Infinite-dimensional vector spaces of sequence 5
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Let
K
=
Q
,
R
,
C
{\displaystyle \mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }
(
p
n
)
n
∈
N
{\displaystyle (p_{n})_{n\in \mathbb {N} }}
monotonic non-increasing sequence with
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 \}}
sequences in
K
{\displaystyle \mathbb {K} }
(
|
a
k
|
p
k
)
k
∈
N
{\displaystyle \left(|a_{k}|^{p_{k}}\right)_{k\in \mathbb {N} }}
For the space
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} }}
Please note, that for all
p
{\displaystyle p}
n
{\displaystyle n}
p
n
{\displaystyle p_{n}}
k
∈
N
{\displaystyle k\in \mathbb {N} }
Impact spaces in normed vector space
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space of polynomial vector
edit
Be
⋅
:
K
×
V
→
V
{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}
⋅
:
K
×
V
→
V
{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}
⋅
:
K
×
V
→
V
{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}
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\}}
⋅
:
K
×
V
→
V
{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}
For a special
(
λ
,
v
)
↦
λ
⋅
v
{\textstyle (\lambda ,v)\mapsto \lambda \cdot v}
(
λ
,
v
)
↦
λ
⋅
v
{\textstyle (\lambda ,v)\mapsto \lambda \cdot v}
(
λ
,
v
)
↦
λ
⋅
v
{\textstyle (\lambda ,v)\mapsto \lambda \cdot v}
(
λ
,
v
)
↦
λ
⋅
v
{\textstyle (\lambda ,v)\mapsto \lambda \cdot v}
Binary operations and functions on vector spaces of sequences 4
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The binary operations and functions on vector spaces of sequences are defined component-wise, analog to addition and scalar multiplication on the vector spacee
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} }
The binary function of scalar multiplication is defined by the multiplication of the components of the sequence with the scalar
λ
∈
K
{\displaystyle \lambda \in \mathbb {K} }
⋅
:
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} }
Consider the set of real numbers
R
{\displaystyle \mathbb {R} }
Q
{\displaystyle \mathbb {Q} }
(
R
,
+
,
⋅
,
Q
)
{\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}
Q
{\displaystyle \mathbb {Q} }
Prove, that the vector
v
1
=
3
{\displaystyle v_{1}=3}
v
2
=
3
{\displaystyle v_{2}={\sqrt {3}}}
linear subspace
U
1
{\displaystyle U_{1}}
Q
{\displaystyle \mathbb {Q} }
(
R
,
+
,
⋅
,
Q
)
{\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}
U
1
∩
U
2
{\displaystyle U_{1}\cap U_{2}}
U
2
:=
5
Q
{\displaystyle U_{2}:={\sqrt {5}}\mathbb {Q} }
0
∈
R
{\displaystyle 0\in \mathbb {R} }
Analyse the subset property of the following vector space of sequences and consider property of convergence of series, which are generated by the sequences with:
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\}}
Identify the subset property between
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)}
![{\textstyle {\mathcal {C}}([a,b],\mathbb {K} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab872fdc352c32b8000a1ff8dcb0b33618d6f65)
![{\textstyle [a.b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eda9b2b1236117cfabf3ecf6085e52579694e191)
![{\textstyle {\mathcal {C}}([a,b],\mathbb {Q} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/701e9696c7cb65de4e6e128cc09ef4d033fdad3f)
![{\textstyle {\mathcal {C}}([a,b],\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9e9e020b3da312fc0e0db761a5fb085942a814)
![{\textstyle {\mathcal {C}}([a,b],\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e23a84bb71309c071bcb762d6cb0a87235c0409c)
![{\textstyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b494d0964a68e4a9a124ee1b6e512843a1b14a)
![{\textstyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108)
![{\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)
