In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well.
Classical Banach spaces
According to Diestel (1984, Chapter VII), the classical Banach spaces are those defined by Dunford & Schwartz (1958), which is the source for the following table.
Glossary of symbols for the table below:
- F {\displaystyle \mathbb {F} } denotes the field of real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .}
- K {\displaystyle K} is a compact Hausdorff space.
- p , q ∈ R {\displaystyle p,q\in \mathbb {R} } are real numbers with 1 < p , q < ∞ {\displaystyle 1<p,q<\infty } that are Hölder conjugates, meaning that they satisfy 1 q + 1 p = 1 {\displaystyle {\frac {1}{q}}+{\frac {1}{p}}=1} and thus also q = p p − 1 . {\displaystyle q={\frac {p}{p-1}}.}
- Σ {\displaystyle \Sigma } is a σ {\displaystyle \sigma } -algebra of sets.
- Ξ {\displaystyle \Xi } is an algebra of sets (for spaces only requiring finite additivity, such as the ba space).
- μ {\displaystyle \mu } is a measure with variation | μ | . {\displaystyle |\mu |.} A positive measure is a real-valued positive set function defined on a σ {\displaystyle \sigma } -algebra which is countably additive.
| 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 ∈ Ξ | μ | ( S ) {\displaystyle =\sup \nolimits _{S\in \Xi }|\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 )} if μ {\displaystyle \mu } is σ {\displaystyle \sigma } -finite. |
| 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])} is the 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])} consists of BV ( [ a , b ] ) {\displaystyle \operatorname {BV} ([a,b])} functions such that 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]),} essentially by Taylor's theorem. |
Banach spaces in other areas of analysis
- The Asplund spaces
- The Hardy spaces
- The space BMO {\displaystyle \operatorname {BMO} } of functions of bounded mean oscillation
- The space of functions of bounded variation
- Sobolev spaces
- The Birnbaum–Orlicz spaces L A ( μ ) . {\displaystyle L^{A}(\mu ).}
- Hölder spaces C k ( Ω ) . {\displaystyle C^{k}(\Omega ).}
- Lorentz space
- ba space
Banach spaces serving as counterexamples
- James' space, a Banach space that has a Schauder basis, but has no unconditional Schauder Basis. Also, James' space is isometrically isomorphic to its double dual, but fails to be reflexive.
- Tsirelson space, a reflexive Banach space in which neither ℓ p {\displaystyle \ell ^{p}} nor c 0 {\displaystyle c_{0}} can be embedded.
- W.T. Gowers construction of a space X {\displaystyle X} that is isomorphic to X ⊕ X ⊕ X {\displaystyle X\oplus X\oplus X} but not X ⊕ X {\displaystyle X\oplus X} serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem1
See also
- List of mathematical spaces – Mathematical set with some added structurePages displaying short descriptions of redirect targets
- List of topologies – List of concrete topologies and topological spaces
- Minkowski distance – Vector distance using pth powers
Notes
- Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5.
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
References
W.T. Gowers, "A solution to the Schroeder–Bernstein problem for Banach spaces", Bulletin of the London Mathematical Society, 28 (1996) pp. 297–304. ↩