List of Banach spaces

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 spacesEdit

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 [a,b]. 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  
np nq Yes Yes  
n n1 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 < ∞
L1(μ) L(μ) No ?   If the measure μ on S is sigma-finite
L(μ)   No ?   where  
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[a,b] rca([a,b]) No No   Isomorphic to Rn ⊕ C([a,b]), essentially by Taylor's theorem.

Banach spaces in other areas of analysisEdit

Banach spaces serving as counterexamplesEdit

NotesEdit

  1. ^ W.T. Gowers, "A solution to the Schroeder–Bernstein problem for Banach spaces", Bulletin of the London Mathematical Society, 28 (1996) pp. 297–304.

ReferencesEdit

  • 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.