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 σalgebra 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 realvalued positive set function defined on a σalgebra which is countably additive.
Classical Banach spaces  
Dual space  Reflexive  weakly complete  Norm  Notes  

Kn  Kn  Yes  Yes  
ℓn_{p}  ℓn_{q}  Yes  Yes  
ℓn_{∞}  ℓn_{1}  Yes  Yes  
ℓ_{p}  ℓ_{q}  Yes  Yes  1 < p < ∞  
ℓ_{1}  ℓ_{∞}  No  Yes  
ℓ_{∞}  ba  No  No  
c  ℓ_{1}  No  No  
c_{0}  ℓ_{1}  No  No  Isomorphic but not isometric to c.  
bv  ℓ_{1} + K  No  Yes  
bv_{0}  ℓ_{1}  No  Yes  
bs  ba  No  No  Isometrically isomorphic to ℓ_{∞}.  
cs  ℓ_{1}  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  V_{f}(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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