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.
Here K denotes the field of real numbers or complex numbers and I is a closed and bounded interval . The number p is a real number with 1 < p < ∞, and q is its Hölder conjugate (also with 1 < q < ∞), so that the next equation holds:
and thus
The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets (for spaces only requiring finite additivity, such as the ba space). The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive.
| Classical Banach spaces | |||||
| Dual space | Reflexive | weakly complete | Norm | Notes | |
|---|---|---|---|---|---|
| Kn | Kn | Yes | Yes | ||
| ℓnp | ℓnq | Yes | Yes | ||
| ℓn∞ | ℓn1 | Yes | Yes | ||
| ℓp | ℓq | Yes | Yes | 1 < p < ∞ | |
| ℓ1 | ℓ∞ | No | Yes | ||
| ℓ∞ | ba | No | No | ||
| c | ℓ1 | No | No | ||
| c0 | ℓ1 | No | No | Isomorphic but not isometric to c. | |
| bv | ℓ1 + K | No | Yes | ||
| bv0 | ℓ1 | No | Yes | ||
| bs | ba | No | No | Isometrically isomorphic to ℓ∞. | |
| cs | ℓ1 | No | No | Isometrically isomorphic to c. | |
| B(X, Ξ) | ba(Ξ) | No | No | ||
| C(X) | rca(X) | No | No | X is a compact Hausdorff space. | |
| ba(Ξ) | ? | No | Yes |
(variation of a measure) |
|
| ca(Σ) | ? | No | Yes | ||
| rca(Σ) | ? | No | Yes | ||
| Lp(μ) | Lq(μ) | Yes | Yes | 1 < p < ∞ | |
| BV(I) | ? | No | Yes | Vf(I) is the total variation of f. | |
| NBV(I) | ? | No | Yes | NBV(I) consists of BV functions such that . | |
| AC(I) | K+L∞(I) | No | Yes | Isomorphic to the Sobolev space W1,1(I). | |
| Cn | rca | No | No | Isomorphic to Rn ⊕ C, essentially by Taylor's theorem. | |
Read more about this topic: List Of Banach Spaces
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