Classical Banach spaces
Dual space Reflexive weakly sequentially complete Norm Notes
F
n
{\displaystyle \mathbb {F} ^{n}}
F
n
{\displaystyle \mathbb {F} ^{n}}
Yes
Yes
‖
x
‖
2
{\displaystyle \|x\|_{2}}
=
(
∑
i
=
1
n
|
x
i
|
2
)
1
/
2
{\displaystyle =\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{1/2}}
Euclidean space
ℓ
p
n
{\displaystyle \ell _{p}^{n}}
ℓ
q
n
{\displaystyle \ell _{q}^{n}}
Yes
Yes
‖
x
‖
p
{\displaystyle \|x\|_{p}}
=
(
∑
i
=
1
n
|
x
i
|
p
)
1
p
{\displaystyle =\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}
ℓ
∞
n
{\displaystyle \ell _{\infty }^{n}}
ℓ
1
n
{\displaystyle \ell _{1}^{n}}
Yes
Yes
‖
x
‖
∞
{\displaystyle \|x\|_{\infty }}
=
max
1
≤
i
≤
n
|
x
i
|
{\displaystyle =\max \nolimits _{1\leq i\leq n}|x_{i}|}
ℓ
p
{\displaystyle \ell ^{p}}
ℓ
q
{\displaystyle \ell ^{q}}
Yes
Yes
‖
x
‖
p
{\displaystyle \|x\|_{p}}
=
(
∑
i
=
1
∞
|
x
i
|
p
)
1
p
{\displaystyle =\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{\frac {1}{p}}}
ℓ
1
{\displaystyle \ell ^{1}}
ℓ
∞
{\displaystyle \ell ^{\infty }}
No
Yes
‖
x
‖
1
{\displaystyle \|x\|_{1}}
=
∑
i
=
1
∞
|
x
i
|
{\displaystyle =\sum _{i=1}^{\infty }\left|x_{i}\right|}
ℓ
∞
{\displaystyle \ell ^{\infty }}
ba
{\displaystyle \operatorname {ba} }
No
No
‖
x
‖
∞
{\displaystyle \|x\|_{\infty }}
=
sup
i
|
x
i
|
{\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}
c
{\displaystyle \operatorname {c} }
ℓ
1
{\displaystyle \ell ^{1}}
No
No
‖
x
‖
∞
{\displaystyle \|x\|_{\infty }}
=
sup
i
|
x
i
|
{\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}
c
0
{\displaystyle c_{0}}
ℓ
1
{\displaystyle \ell ^{1}}
No
No
‖
x
‖
∞
{\displaystyle \|x\|_{\infty }}
=
sup
i
|
x
i
|
{\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}
Isomorphic but not isometric to
c
.
{\displaystyle c.}
bv
{\displaystyle \operatorname {bv} }
ℓ
∞
{\displaystyle \ell ^{\infty }}
No
Yes
‖
x
‖
b
v
{\displaystyle \|x\|_{bv}}
=
|
x
1
|
+
∑
i
=
1
∞
|
x
i
+
1
−
x
i
|
{\displaystyle =\left|x_{1}\right|+\sum _{i=1}^{\infty }\left|x_{i+1}-x_{i}\right|}
Isometrically isomorphic to
ℓ
1
.
{\displaystyle \ell ^{1}.}
bv
0
{\displaystyle \operatorname {bv} _{0}}
ℓ
∞
{\displaystyle \ell ^{\infty }}
No
Yes
‖
x
‖
b
v
0
{\displaystyle \|x\|_{bv_{0}}}
=
∑
i
=
1
∞
|
x
i
+
1
−
x
i
|
{\displaystyle =\sum _{i=1}^{\infty }\left|x_{i+1}-x_{i}\right|}
Isometrically isomorphic to
ℓ
1
.
{\displaystyle \ell ^{1}.}
bs
{\displaystyle \operatorname {bs} }
ba
{\displaystyle \operatorname {ba} }
No
No
‖
x
‖
b
s
{\displaystyle \|x\|_{bs}}
=
sup
n
|
∑
i
=
1
n
x
i
|
{\displaystyle =\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}
Isometrically isomorphic to
ℓ
∞
.
{\displaystyle \ell ^{\infty }.}
cs
{\displaystyle \operatorname {cs} }
ℓ
1
{\displaystyle \ell ^{1}}
No
No
‖
x
‖
b
s
{\displaystyle \|x\|_{bs}}
=
sup
n
|
∑
i
=
1
n
x
i
|
{\displaystyle =\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}
Isometrically isomorphic to
c
.
{\displaystyle c.}
B
(
K
,
Ξ
)
{\displaystyle B(K,\Xi )}
ba
(
Ξ
)
{\displaystyle \operatorname {ba} (\Xi )}
No
No
‖
f
‖
B
{\displaystyle \|f\|_{B}}
=
sup
k
∈
K
|
f
(
k
)
|
{\displaystyle =\sup \nolimits _{k\in K}|f(k)|}
C
(
K
)
{\displaystyle C(K)}
rca
(
K
)
{\displaystyle \operatorname {rca} (K)}
No
No
‖
x
‖
C
(
K
)
{\displaystyle \|x\|_{C(K)}}
=
max
k
∈
K
|
f
(
k
)
|
{\displaystyle =\max \nolimits _{k\in K}|f(k)|}
ba
(
Ξ
)
{\displaystyle \operatorname {ba} (\Xi )}
?
No
Yes
‖
μ
‖
b
a
{\displaystyle \|\mu \|_{ba}}
=
sup
S
∈
Σ
|
μ
|
(
S
)
{\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}
ca
(
Σ
)
{\displaystyle \operatorname {ca} (\Sigma )}
?
No
Yes
‖
μ
‖
b
a
{\displaystyle \|\mu \|_{ba}}
=
sup
S
∈
Σ
|
μ
|
(
S
)
{\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}
A closed subspace of
ba
(
Σ
)
.
{\displaystyle \operatorname {ba} (\Sigma ).}
rca
(
Σ
)
{\displaystyle \operatorname {rca} (\Sigma )}
?
No
Yes
‖
μ
‖
b
a
{\displaystyle \|\mu \|_{ba}}
=
sup
S
∈
Σ
|
μ
|
(
S
)
{\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}
A closed subspace of
ca
(
Σ
)
.
{\displaystyle \operatorname {ca} (\Sigma ).}
L
p
(
μ
)
{\displaystyle L^{p}(\mu )}
L
q
(
μ
)
{\displaystyle L^{q}(\mu )}
Yes
Yes
‖
f
‖
p
{\displaystyle \|f\|_{p}}
=
(
∫
|
f
|
p
d
μ
)
1
p
{\displaystyle =\left(\int |f|^{p}\,d\mu \right)^{\frac {1}{p}}}
L
1
(
μ
)
{\displaystyle L^{1}(\mu )}
L
∞
(
μ
)
{\displaystyle L^{\infty }(\mu )}
No
Yes
‖
f
‖
1
{\displaystyle \|f\|_{1}}
=
∫
|
f
|
d
μ
{\displaystyle =\int |f|\,d\mu }
The dual is
L
∞
(
μ
)
{\displaystyle L^{\infty }(\mu )}
μ
{\displaystyle \mu }
σ
{\displaystyle \sigma }
BV
(
[
a
,
b
]
)
{\displaystyle \operatorname {BV} ([a,b])}
?
No
Yes
‖
f
‖
B
V
{\displaystyle \|f\|_{BV}}
=
V
f
(
[
a
,
b
]
)
+
lim
x
→
a
+
f
(
x
)
{\displaystyle =V_{f}([a,b])+\lim \nolimits _{x\to a^{+}}f(x)}
V
f
(
[
a
,
b
]
)
{\displaystyle V_{f}([a,b])}
total variation of
f
{\displaystyle f}
NBV
(
[
a
,
b
]
)
{\displaystyle \operatorname {NBV} ([a,b])}
?
No
Yes
‖
f
‖
B
V
{\displaystyle \|f\|_{BV}}
=
V
f
(
[
a
,
b
]
)
{\displaystyle =V_{f}([a,b])}
NBV
(
[
a
,
b
]
)
{\displaystyle \operatorname {NBV} ([a,b])}
BV
(
[
a
,
b
]
)
{\displaystyle \operatorname {BV} ([a,b])}
lim
x
→
a
+
f
(
x
)
=
0
{\displaystyle \lim \nolimits _{x\to a^{+}}f(x)=0}
AC
(
[
a
,
b
]
)
{\displaystyle \operatorname {AC} ([a,b])}
F
+
L
∞
(
[
a
,
b
]
)
{\displaystyle \mathbb {F} +L^{\infty }([a,b])}
No
Yes
‖
f
‖
B
V
{\displaystyle \|f\|_{BV}}
=
V
f
(
[
a
,
b
]
)
+
lim
x
→
a
+
f
(
x
)
{\displaystyle =V_{f}([a,b])+\lim \nolimits _{x\to a^{+}}f(x)}
Isomorphic to the Sobolev space
W
1
,
1
(
[
a
,
b
]
)
.
{\displaystyle W^{1,1}([a,b]).}
C
n
(
[
a
,
b
]
)
{\displaystyle C^{n}([a,b])}
rca
(
[
a
,
b
]
)
{\displaystyle \operatorname {rca} ([a,b])}
No
No
‖
f
‖
{\displaystyle \|f\|}
=
∑
i
=
0
n
sup
x
∈
[
a
,
b
]
|
f
(
i
)
(
x
)
|
{\displaystyle =\sum _{i=0}^{n}\sup \nolimits _{x\in [a,b]}\left|f^{(i)}(x)\right|}
Isomorphic to
R
n
⊕
C
(
[
a
,
b
]
)
,
{\displaystyle \mathbb {R} ^{n}\oplus C([a,b]),}
Taylor's theorem .
![{\displaystyle \operatorname {BV} ([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f6474e123b8d06d5989e779c17b9084f2ba8314)
![{\displaystyle =V_{f}([a,b])+\lim \nolimits _{x\to a^{+}}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25115dbc151a6a6ca22cd714e3c0a588ae8c97ab)
![{\displaystyle V_{f}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a55199983d8408d1891b33b485aa49b3b6cd7fc)
![{\displaystyle \operatorname {NBV} ([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97eacd1b77125924adb034d279f23ccb1aae4cfc)
![{\displaystyle =V_{f}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3978b950e1c45a40b89de47e166dff3e3f640f90)
![{\displaystyle \operatorname {AC} ([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1215f6bb4f5dcce36b275a1038200a2da63ffc8)
![{\displaystyle \mathbb {F} +L^{\infty }([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef63fd9a8ef0c7df601ba2aa141815ea86073da)
![{\displaystyle W^{1,1}([a,b]).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a982993a7010fe121285b640c096068e79e74874)
![{\displaystyle C^{n}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5f2c81e52a668fa74a30946eac00229b1d642f)
![{\displaystyle \operatorname {rca} ([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8788ca02e303b567e9d47a44b0fd48a574ddbfb)
![{\displaystyle =\sum _{i=0}^{n}\sup \nolimits _{x\in [a,b]}\left|f^{(i)}(x)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc9f7a9abc638e6fe431d6f36760dbd074b3019)
![{\displaystyle \mathbb {R} ^{n}\oplus C([a,b]),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e48ac47f9406ef46b3c762f19d0d246b54425aa5)
