List of Banach Spaces - 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.

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
p q Yes Yes 1 < p < ∞
1 No Yes
ba No No
c 1 No No
c0 1 No No Isomorphic but not isometric to c.
bv 1 + K No Yes
bv0 1 No Yes
bs ba No No Isometrically isomorphic to ℓ.
cs 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 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.


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